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Final Answers
© 2000-2017   Gérard P. Michon, Ph.D.

Topology  101

Point set topology is a disease
from which future generations will
 regard themselves as having recovered
J. Henri Poincaré  (1854-1912) 

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  • Complete metric space:  All Cauchy sequences are convergent.
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Related Links  (Outside this Site)

The Beginning of Topology in the US & the Moore School  by  F. Burton Jones
Topology Atlas  |  Q&A  by  Henno Brandsma  at York University (Toronto).
The Topological Zoo   ( Geometry Center  of the University of Minnesota ).
A History of Topology  by  J.J. O'Connor  and  E.F. Robertson  (MacTutor)
Knots and Their Polynomials  by  Tony Phillips  (AMS).
Knot Theory  by Nicholas B. Tufillaro, Jeremiah Reilly, and Tyler Abbott.
Euler Characteristic versus Homotopy Cardinality  by  John C. Baez.
Hadwiger's theorem:  What can we measure?  [ 2 ]  by  Dan Piponi (sigfpe)
Interactive Real Analysis  (topology)  by  Bert G. Wachsmuth.
Topology for Physicists  by  Alexander Abanov  (Stony Brook University)
Integration Workshop 2003  by  Douglas Ulmer  (University of Arizona)
What is Boy's Surface?  by  Rob Kirby   (Notices of the AMS, Nov. 2007)
Topology Notes  by  Damon J. Wischik  (Cambridge LDQ Course)
MathWorld :     Fiber Bundle  by Todd Rowland
Wikipedia :     Topology   |   Compact space   |   Product topology   |   Winding number   |   Homotopy Principle   |   Poincaré Duality   |   Complex Measure   |   Totally Disconnected   |   Arrangement of Hyperplanes   |   Long line   |   Polish space



Topology  101

 Maurice Frechet 
 1878-1973 (2007-11-02)   Metric Spaces   (Maurice Fréchet, 1906)
Distance entails a particular topological structure.

Many topological notions  (continuity, connectedness, etc.)  were first introduced in the context of a  metric space,  where a  distance  d  is defined which is endowed with the following axiomatic properties:

  • d(x,y)  is a nonnegative real number  (called  distance  from  x  to  y).
  • d(x,y)  =  d(y,x).
  • d(x,z)  never exceeds  d(x,y) + d(y,z)   (the  triangular inequality ).
  • d(x,x)  =  0
  • d(x,y)  is zero only if  x = y  (or else,  d  is called a  semidistance).

From a modern viewpoint, topological properties are based on the concept of an  open set,  as discussed in the next article.  In a metric space, an open set is a set which contains an open ball centered on  every  point in it.

The  open ball  of center  C  and (positive) radius  R  is the set of all points whose distance to  C  is less than  R.

 Bertus Brouwer 
 (1881-1966) (2007-11-02)   Topological Spaces   (Brouwer, 1913)
Defining a topology is singling out some subsets as  open.

A set  E  is said to be a  topological space  when it possesses a specific  topology.  Formally, a  topology  is simply a particular collection of subsets, called  open sets  verifying the following axiomatic properties  (L.E.J. Brouwer, 1913).

  • The empty set  (Æ)  is open.
  • The whole set  (E)  is open.
  • Any union of open sets is open.
  • Any intersection of  finitely many  open sets is open.

In particular, it can be checked that those axioms are verified for the  open sets  defined as above in the special case of metric spaces.

The  trivial  (or  indiscrete)  topology is  { Æ, E } ;  only  Æ  and  E  are open  (in French, this is called  topologie grossière ).

At the other extreme, with the  discrete  topology, every set is open.

Neighborhood :   A  neighborhood  of  X  (X may be either a point or a set)  is a set which contains some open set containing  X.

Open neighborhood :   A neighborhood which is an open set...  Thus, an open neighborhood of  X  is simply an open set containing  X.  Open neighborhoods are the only type of neighborhoods some authors will consider.  There are good historical reasons for that viewpoint, but the modern nomenclature has now freed itself from that constraint, so we may speak freely about interesting things like closed neighborhoods or compact neighborhoods...

Interior :   The  interior  Å of a set A is the union of all open sets it contains.  (That's the  largest  open set contained in A.)

Limit-point :   A point  a  of a topological space  E  is said to be a  limit-point  of a subset A when every [open] neighborhood of  a  intersects A in a least one point besides A.  This concept was introduced by Cantor in 1872  (in the special case of the metric space of real numbers).

(2014-11-26)   Limits in a Topological Space
Generalization of the metric notion of a limit.

An element  a  of a topological space is said to be the limit of a sequence  a  of other points when, for any given integer N, there is a neighborhood of  a  which contains every  a  for  n ≥ N.

A sequence is said to be  convergent  if it has a limit.  In some  coarse  topological spaces,  a convergent sequence may have more than one limit.  However,  such spaces are infrequently considered in practice  (except, possibly, as a source of counterexamples for honing the definitions of general topological concepts).  Usually, we only bother with  Hausdorff spaces,  where the limit of a sequence,  if it exists,  is necessarily  unique.

(2013-01-04)   Basis of a Topological Space
The open sets are all the unions of sets from a  topological base.

There's at least one such basis:  the topology itself  (the set of all open sets).

The smallest possible cardinality of a basis is the  weight  of the topology.

Subbasis :

A set  B  of open sets is a  subbasis  of the topology if no lesser topology exists where all the elements of  B  are open sets.

Equivalently,  B  is a subbasis of the topology if and only if a basis of the topology is formed by the empty set and all finite intersections of sets of  B  (including the nullary intersection, which is equal to the entire space).

Local Basis :

local basis  (or  neighborhood basis )  at point  x  is a family of open sets of which every neighborhood of  x  contains a member.

First-Countable Spaces :  (Hausdorff, 1914)

first-countable space  is a topological space for which there's a countable  local basis  at every point.

Every metric space is first-countable  (the open balls of rational radius centered on x form a countable  local basis  at point x).

Second-Countable Spaces :

second-countable space  is a topological space for which there's a  countable basis.  A second-countable space is clearly first-countable.

A second-countable space is separable.  It's also Lindelöf  (every open cover contains a countable subcover).  In the particular case of  metric spaces,  the three properties  (second-countability, separability and Lindelöf)  are equivalent.

A countable product of second-countable spaces is second-countable.  However an uncountable such product may not even be first-countable.

Wikipedia :   Topological base   |   Axioms of countability
First-countable space   |   Second-countable space (Completely separable space)

(2007-11-02)   Closed Sets
A subset is closed when its complement is open

In a  topological space  (as defined above)  a set is said to be  closed  when its complement is  open  (the complement of a subset F of a set E is the subset of E consisting of all the elements of E which are not in F).

Clearly, a  topology  on a set  E  could be specified by indicating which of its subsets are  closed.  If such a viewpoint is adopted, the  closed sets  must simply verify the following axiomatic properties:

  • The whole set  (E)  is closed.
  • The empty set  (Æ)  is closed.
  • Any intersection of closed sets is closed.
  • Any union of  finitely many  closed sets is closed.

The equivalence of those properties with the axiomatic properties previously stated for open sets is based on the fact that the complement of an intersection is the union of the complements, whereas the complement of a union is the intersection of the complements  (de Morgan's Laws).

One particular topology which is best defined this way is the so-called  cofinite  topology, for which the only  closed sets  are  Æ,  E  and all its  finite  subsets.

Closure :   The  closure  (French: adhérenceA of a set A is the intersection of all closed sets which contain it.  (It's the  smallest  closed set containing A.)

Border (or boundary) :   The  border  (or  boundaryA  of a set  A  is the intersection of the closure of A and the complement of the interior of A.  ( A is a  closed set,  because it's the intersection of two closed sets.)

Dense subset :   A set is said to be  dense  in a topological space when its  closure  is equal to the entire space.

Nowhere-dense set :   A set is said to be  nowhere dense  when its  closure  has empty  interior.

Meager set :   Countable union of nowhere dense sets.

Almost-open subset :   A subset is said to be  almost open  if it has the property of Baire  (i.e.,  its symmetrical difference with some open set is a meager set).

Separability :   A  topological space  is said to be  separable  if it's the closure of a countable subset  (equivalently, if a countable set is  dense in it).  For example, the real line  R  is separable because it's the closure of the set of rational numbers  Q ,  which is indeed countable.

(2007-12-06)   Subspace  F  of a topological space  E.
Open sets of  F  are intersections with  F  of open sets of  E.

subspace  F  of a topological space  E  is a subset of  E  endowed with the so-called  induced  topology where an open set of  F  is defined to be the intersection with  F  of an open set of  E.

Equivalently, a closed set of  F  is  defined  to be the intersection with  F  of a closed set of  E.

A subspace  F  is always both open and closed in itself, but it need not be either open or closed in the whole space  E.  (F  could be  any  subset of  E).

(2007-11-09)   Optional  separation axioms   ( Trennungaxiome ).
The fewer the open sets, the  coarser the topology.

The basic structure of a topological space is not sufficient to support some general statements that require various assumptions about how a topology distinguishes between points.  Historically, some of the following "separation axioms" were once considered for inclusion in the general definition of a  topological space.  Mercifully, none of them have been retained in that general capacity.  Each of them just denotes a particular class of topologies which is specified whenever the corresponding properties are needed.  The standard characterizations of all separation axioms are of the form: 

Two things separated in a weak sense are separated in a stronger sense.

A subscripted  T  denotes a choice of a separation axiom within the most commonly used hierarchy, tabulated below, where  Ti  implies  Tj  if  i > j.

The letter  T  is for  Trennung,  a German word meaning "separation".  (Separability is a different concept,  in English and French, at least.)
Most commonly, a topological space is said to be "separated" when is verifies the "Hausdorff condition"  (i.e., the separation axiom  T).  Such a separated topological space is best called a Hausdorff space.  In a Hausdorff space, a convergent sequence has a unique limit.

Types of topologies, from coarser to finer separation   (T = Trennung axiom)
TNameDefinitions / Comments
T0 Kolmogorov For any pair of distinct points {x,y} there's an open set containing x but not y, or y but not x.
T1Fréchet For any couple of distinct points  (x,y)  there's an open set containing x but not y.
(Equivalently:  All finite sets are closed.)
Two distinct points
always have disjoint neighborhoods
(Equivalently:  Any pair of points is disconnected.)
T Urysohn Two distinct points always have
disjoint closed neighborhoods.
RegularA closed set and a point outside of it
always have disjoint neighborhoods.
T3Regular HausdorffRegular  T0  space.
Completely RegularA closed set and a point outside of it are
always separated by a function.
T Tychonoff Completely regular  T0  space.
NormalTwo disjoint closed sets
always have disjoint neighborhoods.
T4Normal HausdorffNormal  T1  space.
Completely normalTwo disjoint sets
always have disjoint neighborhoods.
T5Completely Normal HausdorffCompletely normal  T1  space.
Perfectly normalTwo disjoint sets are always
precisely separated by a function.
T6Perfectly normal HausdorffPerfectly normal  T1  space.

The trivial  or  indiscrete  topology  (where every nonempty open set contains everything)  is clearly the  coarsest  possible topology.  It doesn't verify any of the above  separation axioms  (except if the entire space is empty or contain just a single point).  At the other extreme, the discrete topology  (where every subset is open)  satisfies every conceivable separation axiom.  Neither of those two extreme topologies is very useful, except to provide didactic examples or counterexamples.

On any  infinite  space,  the cofinite  topology  (where the  closed  proper subsets are just the  finite  proper subsets)  verifies  T1  but nothing stronger.  (HINT:  The intersection of two nonempty open sets is infinite.)

Any  metric space  does  satisfy  all  of the above separation axioms.

Proofs  (or exposed tautologies) :

  • T1   =>   T0     Trivially.   QED
  • T1   <=>   { All finite sets are closed. } :
    Let  F  be a finite subset of a  T1  space  E.  Consider a given point y of F.  For any other point x of E, there's an open set containing x but not y.  The union of all such open sets is an open set containing every point of E except y; it's thus the complement of {y}.  Having an open complement, {y} is closed.  So is F, as a finite union of closed sets.  Conversely,  assume that every finite subset of E is closed.  Then, for any ordered pair of distinct points  (x,y), the complement of {y} is an open set containing x but not y.  Therefore, E obeys T1  QED
  • T2   =>   T1   :
    If distinct points x and y have disjoint neighborhoods, we may pick an open set in such a neighborhood of y which doesn't contain x because it doesn't intersect the relevant neighborhood of x.  QED
  • T2   <=>   { A pair of distinct points is always a disconnected set. } :
    Consider two distinct points x and y in a space obeying  T.  As they have disjoint neighborhoods, we have two disjoint open sets such that one contains x but not y and the other contains x but not y.  Thus, {x} and {y} are disconnected from each other. 
    Conversely,  assume that two distinct points x and y are always disconnected from each other.  This means that we have two disjoint open sets which contain x and y respectively.  As those are disjoint neighborhood of x and y,  T2  holds.  QED

 Come back later, we're
 still working on this one...

Wikipedia :   Separation axioms   |   History of the separation axioms   |   Topological properties
Felix Hausdorff (1868-1942)

(2007-11-09)   Compactness; Heine-Borel criterion   (Fréchet, 1906)
A set is compact when any  open cover  includes a  finite  subcover.  (An  open cover  is a union of open sets containing the prescribed set.)

A set is said to be  relatively compact  when its  closure  is compact.

Two closely related notions  (which are  equivalent to compactness  in the case of metric or metrizable spaces)  are  sequential compactness  (a set is sequentially compact when any sequence in it contains a convergent subsequence)  and  limit-point compactness  (the property possessed by a set which contains a limit-point of every infinite subset).

The  Bolzano-Weierstrass theorem  (Bolzano, 1817)  affirms that  "every bounded sequence of points [in finite-dimensional Euclidean space] has a convergent subsequence".  It's thus equivalent to the Heine-Borel theorem:

The  Heine-Borel theorem  says that, in the Euclidean space  Rn,  a set is compact if and only if it's  closed  and  bounded.

In this, a compact set is defined in the modern sense as "a set for which any open cover contains a finite subcover"  which is known as the Heine-Borel criterion  (as opposed to the aforementioned Bolzano-Weierstrass criterion defining sequential compactness, which may not be equivalent to compactness for non-metrizable topological spaces).

More generally, in any metric space,  a set is  compact  if and only if it's  complete  and  totally bounded.

A subset of a metric space is said to be totally bounded when it can be covered by finitely many balls of radius  r,  for any given radius  r.  In a Euclidean space of infinitely many dimensions, a bounded set  (like a ball of unit radius)  need not be totally bounded.  Actually, a closed ball is compact only in a space of finitely many dimensions.

For any topological space, a closed subset of a compact set is compact.  Also, the intersection of a closed set and a compact set is compact.

In a  Hausdorff space,  a  compact set  is necessarily  closed.

(2012-11-13)   The  extreme-value  theorem   (Bolzano, c.1830)
A continuous real-valued function defined on a compact set is bounded and  attains  both of its extreme values.

This is, arguably, a fundamental theorem in mathematical optimization.

The original theorem  (for functions defined in n-dimensional Euclidean space)  was proved in the 1830s  by the Germanophone Bohemian mathematician  Bernard Bolzano (1781-1848)  whose relevant work  (Functionenlehre)  was only published in 1930.  That theorem was established independently by  Karl Weierstrass (1815-1897)  in 1860.

Bolzano's proof consisted of two steps:  First, he showed that a continuous function over a compact domain had to be bounded.  Second, he proved that the least upper bound and greatest lower bound are attained at some points.  For both steps, Bolzano invoked what's now called the  Bolzano-Weierstrass theorem  ("every bounded sequence contains a convergence subsequence")  which he had established in 1817,  as a lemma in the proof of his version of the  intermediate value theorem  ("a continous function with negative and positive values must vanish at some point").

Nowadays, to establish the theorem for compact sets in any topological space  (not necessarily a metric or metrizable one)  we merely observe that the continuous image of a compact set of points is a compact set of reals, namely a closed bounded set of real numbers.  Thus, there is a minimum and a maximum and both are the images of some points...

A continuous image of a compact set is compact.

Proof :   If  f  is continuous and A is compact, consider  any  open cover of  f (A).  The preimages of the elements of that cover are open sets  (because  f  is continuous)  which cover  A.  Therefore, since  A  is compact, we can select a finite number of those which cover  A.  The images of that selection form a finite subcover of the original cover of  f (A).   QED

The converse isn't true:  Functions that send compacts to compacts aren't necessarily continuous.  For example, any function that takes on only finitely many values has this property  (since finite sets are compact)  but is not necessarily continuous.

Extreme value theorem

(2012-09-21)   Borel Sets.  Borelian Tribe.   (1898)
The  two  definitions of a Borel set are usually equivalent.

s-algebra  or  tribe  (the French term  tribu  was introduced in 1936 by the Bourbakist  René de Possel, 1905-1974)  is a family of sets closed under  countable intersectioncountable union  and  relative complement.

The open sets need not form a tribe.  The  smallest  tribe containing all open sets  (the intersection of all tribes that contain all open sets)  is called the  Borel tribe  or the  Borelian tribe.  Its elements are called  Borel sets  or  Borelians.

Borel sets are thus derived from open sets by countable union, countable intersection and relative complement.

Equivalently, we may start with the closed sets.  The same Borel tribe is also obtained as the smallest tribe containing all closed sets.

Some authors have proposed to define the  Borel tribe  as the smallest tribe containing all the compact sets.  This definition is  usually  equivalent to the classical definition presented above, but for some  pathological  topologies, this ain't so...

Borel set   |   Emile Borel (1871-1956)
Borel Spaces (pdf)  by Sterling K. Berberian, Ph.D. (1955)   University of Texas at Austin   (1986, 1997)

(2007-11-17)   Locally Compact Spaces
Spaces in which every point has a compact neighborhood.

As is demonstrated by the  Heine-Borel Theorem  for metric spaces, compactness and completeness are strongly related but compactness implies an overall limitation which is not present in the purely local concept of completeness.

Traditionally, completeness is only defined for  metric spaces  (because Cauchy sequences are a purely metrical concept).  A loose counterpart of completeness in general topological spaces, must involve some concept of  local compactness.  All the definitions which have ever been proposed are equivalent to the one featured above in the case of Haudorff spaces.

Again, local compactness is a relatively minor  topological  concept which is only loosely related to the very important  metric  concept of  completeness  (which  André Weil  extended to  uniform spaces  in 1937).

Topological Vector Space :

A topological vector space is locally compact iff it's finite-dimensional.

This classical result is due to  Frédéric Riesz  (Riesz Frigyes, in Hungarian).

Riesz' lemma  |  Frigyes Riesz (1880-1956) is the older brother of Marcel Riesz (1886-1969)

(2007-11-02)   Sequence Characterizations
Characterizing a set by metric properties of the sequences in it.

Let  U  be a subset of a  metric space  E.

  • U is  closed  if and only if it contains the limit of all convergent sequences of its own points.
  • U is compact when any sequence of its points has a subsequence which converges in U.
  • U is complete iff any Cauchy sequence of points of U converges in U.

(2007-11-02)   Continuous Functions
A function is continuous  iff  the  inverse image  of any open set is open.

Arguably, one of the original motivations of the entire field of topology was to characterize continuity in very general terms.  This is achieved by the above definition, which looks natural only after years of proper mathematical training... 

Two equivalent statements characterize a  continuous function  f  defined over some subset  D  of one topological space with values in another:

  • The inverse image of any open set is a set which is open in  D.
  • The inverse image of any closed set is a set which is closed in  D.

Recall that a set  "open (resp. closed) in D"  is the intersection with D of an open set  (resp. a closed set).

Continuous functions verify two important properties, respectively known as the  extreme value theorem  and the  intermediate value theorem  (at least, that's the name they have for real functions of a real variable).  Namely:

However, neither statement is characteristic of continuous functions  (i.e.,  each can be satisfied by  some  discontinuous functions as well).

(2012-12-27)   Restricting or extending a  continuous  function.
A continuous function restricted to a  subspace remains continuous.

The restriction of a continuous function to a  topological subspace  is always continuous.  However, there may not always be a way to  extend  a continuous function defined on a subspace to a continous function defined on the whole space...

One example  (butchered by educators with weak topological skills)  pertains to the expression   f (x) = 1/x   (a simple homographic transformation).  This relation does define a  continuous  bijection  f  from  D  to  D  when  D  is one of the following sets.  (Continuity is counterintuitive in the first case!)

  • R*   (the nonzero reals, separated into two connected components).
  • R U {¥}  (reals with single  unsigned  infinity; topology of a circle).
  • C*   (the nonzero complex numbers).
  • C U {¥}  (projective complex line; topology of a sphere).

However, no continous extension of  f  can be defined from  R  (all real numbers, including zero)  to the closed interval  [-¥, +¥]  endowed with the usual topology  (where the positive reals do  not  constitute a neighborhood of negative infinity, and vice-versa).

This is why three different types  of pseudo-numerical infinities are defined  (or ought to be defined)  in  computer algebra systems  (CAS):

  • ¥  =  1/0  :   Unsigned, complex or  algebraic  infinity.
  •   and    :   Signed infinities  (used mostly in  real  analysis).

(2007-11-09)   Product Topology,  Tychonoff Topology  (1926, 1935)
Coarsest topology for which all  projections  are continuous.

Consider the cartesian product  E  of finitely or infinitely many sets:

E   =     Õ   Ei

For each index  i,  there's a  projection function  pi  which transforms  an element  x  of  E  into the corresponding component of  x  in  E.  Formally:

{ x }   =     Õ   { pi (x) }

E  is best endowed with the least topology which makes all such projections continuous.  (Recall that the "topology" is, formally, the collection of all open sets.)  This so-called  product topology  can also be described as consisting of all unions (finite or infinite) of  finite  intersections of sets of the following form:

Õ   Ui     separator where  Ui  is an open subset of  Ei  which is
different from  Ei  in only  finitely many  cases.

If we didn't insist on  U being a proper open set for only finitely many indices, we would obtain a  finer  topology known as the  box topology.

The above  product topology  is often called the Tychonoff topology.  It was discovered by Andrei Nikolaevich Tikhonov (1906-1993) in 1926, before he even graduated...  Arguably, this is the only "correct" topology to consider over a cartesian product of topological spaces.  In particular, it ensures that a map  f  is continuous  if and only if  its components  fi  are continuous.

f (x) }   =     Õ   {  fi (x) }

This desirable theorem would not be true, in general, with the  box topology,  which is  too fine  and makes it much harder for a function to be continuous.  Similarly, Tikhonov proved that his  product topology  makes any product of compact spaces compact.  By comparison,  box topology  looks like a misguided idea  (except for a finite cartesian product, or when almost all components are endowed with the trivial topology, in which cases the two concepts coincide).

Product of Separable Spaces :

A countable product of separable spaces is separable  [ proof ].  So is a separable space raised to the power of the continuum  [ proof ].

Tychonoff's Theorem  (1930, 1935)

The cartesian product of any collection of compact spaces is compact.

This is one of the most important results of general topology.  It helped define the modern concept of compactness based on the  Heine-Borel criterion  (every open cover has a finite subcover).  That definition replaced a definition of compactness, now called  sequential compactness,  based on the  Bolzano-Weierstrass  criterion  (any sequence has a convergent subsequence).  Both definitions are equivalent for metrizable spaces but neither implies the other for [some?] other topological spaces.

For example, the product of an uncountable number of copies of the closed unit interval  fails  to be sequentially compact.

Tychonoff 's theorem  relies on the  Axiom of choice.  In fact,  Tychonoff 's theorem  and the  Axiom of choice  turn out to be equivalent statements.

Cauchy's Mistake  (1821) :

Cauchy  thought that a function of two variables  x  and  y  which is continuous with respect to  x  and with respect to  y  must be continuous with respect to  (x,y).  This ain't so.  The following counterexample was produced in  1870  by  Johannes Thomae (1840-1921).  It has continuous projections but is discontinuous at the (0,0) point.  (HINT:  Consider the line  y = a x )

f (x,0)=0
If  y ¹ 0 ,     f (x,y)= sin ( 4 Arctg x   )

Hewitt-Marczewski-Pondiczery theorem (1944-1947)
Edwin Hewitt (1920-1999)   |   Edward Marczewski (1907-1976)
"E.S. Pondiczery"  was actually a pseudonym for  Ralph_P._Boas, Jr. (1912-1992)

(2007-11-02)   Connected Set
A connected set cannot be split by two disjoint open sets.

By definition :

  • Two sets are said to be  disconnected from each other  if they are respectively contained in two  disjoint  open sets.
  • A set is said to be  disconnected  if it's the union of two  nonempty  parts that are disconnected from each other. 
  • A set is said to be  connected  if it's not disconnected.

To prove that a set  A  is connected, we may show that it can't be contained in the union of of two disjoint open sets  U  and  V  unless one is empty.

A nonempty  topological space  E  is  connected  if and only if it doesn't contain any  clopen  (i.e., both open and closed)  nonempty proper subset.

The empty set is connected.  So is any set containing only one point.

A topological space where there are no other connected sets is said to be  totally disconnected  (in such spaces, there are nonconnected sets with more than one point).  For example, the discrete topology always produces a  totally disconnected  topological space.  More interestingly, the followings spaces are totally disconnected:

On the other hand, with the trivial topology  (the so-called  indiscrete topology)  every set is connected.

The closure of a connected set is connected.

Proof :   If the closure of a set is disconnected, then that closure can be split by two disjoint open sets which also split the set, proving it's disconnected.

Connected Components :

connected component  of a topological space is a maximal connected  nonempty  set  (i.e., a nonempty connected set which isn't contained in any larger connected set).

The convention is thus made that the empty set  isn't   a connected component of anything  (not even itself)  although it's definitely connected.

All the connected components form a unique  partition  of the topological space  (i.e., they're pairwise disjoint and their union is the whole space).  This fundamental property is the reason why we had to rule out the empty set as a connected component  (since elements of a partition are never empty).

An empty collection of sets is a valid partition which corresponds to the connected components of an empty topological space  (which has no connected components, as previously noted).

Similar definitions apply to other restricted notions of connectedness  (path-connectedness and arc-connectedness).

Every connected component must be  closed  (HINT:  its closure is connected).  If there are only finitely many of them, each is also  open  (HINT:  its complement is a finite union of closed sets).  If there are  infinitely many  connected components, they're not necessarily open  (e.g., Cantor set).

Connectedness and Continuity :

The above definition of continuity satisfies the intuitive requirement that a continuous function must transform a connected set into a connected set... 

A continuous image of a connected set is connected.

Proof :   Let  f  be a continuous function.  By definition, it is such that the  inverse image  of any open set is an open set.  We have to prove that the  direct image  f (A)  of a connected set  A  is connected or,  equivalently,  the  contrapositive  statement:  If  f (A)  is disconnected, so is  A.

Well, if  f (A)  is disconnected, we can split it into two nonempty parts respectively contained in two disjoint open sets  U  and  V
Consider the open sets  f  -1 ( U )  and  f  -1 ( V )...

  • Neither has an empty intersection with  A, because  U  and  V  both have at least one element from  f (A).
  • Every element of  A  is in one or the other of those two open sets.
  • Their intersection is empty, because any element of both would need to have its image in both  U  and  V,  which are disjoint.

A  is thus split into nonempty parts by two disjoint open sets.    QED

Note, however, that the converse is false:  There are  discontinuous  functions which transform every connected set into a connected set.  The  following section  provides many such examples in the special case of real functions of a real variable...

(2014-09-01)   Intermediate-Value Theorem
A continuous function from reals to reals maps an interval to an interval.

On the real line, the connected sets are the intervals.  So, that's just a special case of the general result established in the previous section  (a continuous image of a connected set is connected).

By contrast, the intervals of rationals are not connected  (the set of rational numbers is totally disconnected)  and there is no equivalent of the intermediate-value theorem for rationals.

Spelled out in elementary terms, this yields a very useful result which says that, for any  continuous  function of a real variable defined between  a  and  b,  any value  y  between  f (a)  and  f (b)  is equal to  f (x),  for some  x.

A popular formulation used by Bolzano  (for functions on an interval)  is: Continuous functions with positive and negative values vanish somewhere.

The converse isn't true :

The intermediate-value property is  not  a characteristic property of continuous functions:  There are functions which are not continuous for which the property holds.  Such is the case for any discontinuous derivative  f '  of a differentiable function.  For example, we may use:

f (x)   =   x 2  sin ( 1/x )       [ with  f (0)  =  0 ]

Proof :   Since any differentiable function  f  is continuous,  the  extreme value theorem  states that it must reach a minimum  and  a maximum within any interval  [a,b]  on which it is defined.  If  f ' (a)  and  f ' (b)  have opposite (nonzero) signs,  at least one of those extrema is not located at an extremity, so it must be at a point  x  where   f ' (x) = 0   QED

Wikipedia :   Intermediate value theorem

(2012-12-30)   Path Connectedness
A path-connected set is connected.  The converse may not be true.

In a  topological space  X,  a  path  from  a  to  b  is defined to be a continuous function  f  from the closed interval  [0,1]  to  X  such that:

f (0)   =   a       and       f (1)   =   b

A subset  Y  is said to be  path-connected  (or  pathwise connected )  when such a path exists whose image is  contained  in  Y,  for any pair of extremities  {a,b}  in  Y.

Some authors have used the terms  arc-connected  (or  arcwise connected )  for that same concept.  However, this practice is not recommended, since the term is best used for a stronger concept.

A set consisting of a single point is path-connected.  The empty set isn't.  (Thus the empty set is a  trivial  example of a connected set which isn't path connected.)

nontrivial  example of a connected set which isn't path-connected is the  closure of the so-called  topologist's sine curve ;  the planar curve of cartesian equation:

y   =   sin ( 1/x )     for   0  <  x  < 2/p

That closure includes the  segment  at  x = 0,  between  (0,-1)  and  (0,1).

Lemma :  The interval  [0,1]  is connected  (proof by contradiction).

Theorem :  Any path-connected set is connected.

Proof :   Consider two  arbitrary  points  a  and  b  of a path-connected set  Y.  Let  P  be the image of a path joining them within  Y.  If the two extremities were respectively in two disjoint open sets  U  and  V  whose union contained  Y,  then those two open sets would likewise split  P  and prove it to be  disconnected.  Since we know that  P  is connected (as a continuous image of the connected set [0,1] examined in our lemma)  we deduce that  a  and  b  cannot possibly be in two disjoint open sets covering Y.  As this is true of any pair of points of Y, there cannot be two nonempty parts of Y in disjoint open sets covering Y.  Therefore,  Y  is connected.    QED

Lemma :  In a normed space, balls and convex sets are path-connected.
(HINT:  Consider the  path   f (u)  =  (1-u) a  +  u b )

Theorem :  In a normed space, a connected  open  set is path-connected.

Proof :   For any point  a  of a nonempty open set  U,  we may define the following two sets V and W.  Both are open.  (HINTS:  Open balls are path-connected  (lemma).  The union of two arcs sharing an extremity is an arc.)

  • V  consists of all points  b  of  U  for which there is a path from  a  to  b.
  • W  consists of all points  z  of  U  for which there's  no  path from  a  to  z.

U is the union of the two disjoint open sets V and W.  V is nonempty, since it contains  a.  Therefore, if U is connected, W must be empty,  which means that there's a path from  a  to any other point of  U.   QED

(2007-10-31)   Homeomorphisms
Continuous bijections whose inverses are also continuous.

homeomorphism  is simply a  bicontinuous  function  (which is to say that it's continuous and bijective and that its inverse is continuous as well).

An homeomorphism can be construed as an  isomorphism  of the topological structure.  A bijection is an homeomorphism if and only if it transforms any open set into an open set and any closed set into a closed set.

Two topological sets are said to be  homeomorphic  when there's an homeomorphism between them.

(2012-12-30)   Arcs  &  Arc-Connectedness
An  arc  is a topological subspace  homeomorphic  to  [0,1].

An arc is said to  join  its pair of extremities  (defined as the image of {0,1} under any homeomorphism between the arc and the  interval  [0,1] ).

A topological space where any point is joined to any other point by an arc is said to be  arc-connected  or  arcwise connected.

Clearly, an arc-connected space is path-connected  (since bicontinuous functions are continuous).  However, the converse need not be true.

A simple counterexample is a topological space  X  consisting of just two points under the  trivial topology:  That space  X  is  not  arc-connected  because there are no  injections  from [0,1] to X, because [0,1] has more than two elements  (hence no bicontinuous functions between [0,1] and the only pair of points in X).  On the other hand, there's a continuous path from one point of  X  to the other, obtained from a function which is equal to one point of X at zero and the other point elsewhere.  (That function is continuous because the inverse image of the only nonempty open set of X is equal to [0,1],  which is open in itself.)

Mathworld :  Arcwise-connected

(2007-10-31)   Homotopy  &  homotopic functions   (Jordan, 1866)

A homotopy between two continuous functions  f  and  g  from X to Y is a  continuous  function  h  from  X ´ [0,1]  to  Y  such that

"x       h (x,0)   =   f (x)     and     h (x,1)   =   g (x)

If such a homotopy exists, the functions  f  and  g  are said to be  homotopic.

Path-connectedness is homotopy-invariant  by  Damek Davis   (2011-09-19).
Camille Jordan (1838-1922; X1855)

(2007-11-05)   The Fundamental Group  (first  homotopy group )
The homotopy classes of the loops going through a given  base point.

In a  topological space  X,  a  loop  through point  a  is a continuous function  f  from the closed interval  [0,1]  to  X  such that:

f (0)   =   a       and       f (1)   =   a

 Come back later, we're
 still working on this one...

(2007-11-06)   Homotopy Groups
Generalizing the fundamental group to  n-dimensional  hyperloops.

 Come back later, we're
 still working on this one...

(2007-11-06)   Diffeomorphisms
Differentiable maps with differentiable inverses.

 Come back later, we're
 still working on this one...

(2007-10-31)   Homology  &  Cohomology

 Come back later, we're
 still working on this one...

 cylinder (J. T. of Summerville, SC. 2000-11-19)
How many edges (lines) are in a cylinder?

I assume we're talking about a finite cylinder; the "ordinary kind" with two parallel bases, which are usually circular (as opposed, say, to an infinite cylinder with an infinite lateral surface and no bases).

The answer is, of course, that there are two edges, the two circles.

I think you figured this out by yourself and did not need anybody to tell you, so I suppose your real concern is elsewhere...

Leonard Euler Rene Descartes Because you used the term "edges" I suspect you think you've found an exception to the Descartes-Euler formula, which states that "in a polyhedron" the numbers of faces (F), edges (E) and vertices (V) are related by the formula: F-E+V=2.

In a way, you have such a "counterexample": In a cylinder, there are 3 faces (top, bottom, lateral), 2 edges (top and bottom circles) and no vertices, so that F-E+V is 1, not 2! What could be wrong?

Nothing is wrong if things are precisely stated. Edges and faces are allowed to be curved, but the Descartes-Euler formula has 3 restrictions, namely:

  1. It only applies to a (polyhedral) surface which is topologically "like" a sphere (imagine making the polyhedron out of flexible plastic and blowing air into it, and you'll see what I mean). Your cylinder does qualify (a torus would not).
  2. It only applies if all faces are "like" an open disk. The top and bottom faces of your cylinder do qualify, but the lateral face does not.
  3. It only applies if all edges are "like" an open line segment. Neither of your circular edges qualifies.

There are two ways to fix the situation. The first one is to introduce new edges and vertices artificially to meet the above 3 conditions.  For example, put a new vertex on the top edge and on the bottom edge. This satisfies condition (3), since a circle minus a point is "like" an open line segment.  The remaining problem is condition (2); the lateral face is not "like" an open disk (or square, same thing).  To make it so, "cut" it by introducing a regular edge between your two new vertices.  Now that all 3 conditions are met, what do we have? 3 faces, 3 edges and 2 vertices.  Since 3-3+2 is indeed 2, the Descartes-Euler formula does hold.

The better way to fix the formula does not involve introducing unnecessary edges or vertices.  It involves the so-called Euler characteristic, often denoted c (chi):

The Euler Characteristic  c  ( chi )

The fundamental properties of c (chi) may be summarized as follows :

  1. Any set with a single element has a c of 1 :   "x,  c ( {x} )  =  1
  2. c is additive:  For two disjoint sets E and F,  c(EÈF) = c(E) + c(F)
  3. If E is homeomorphic to F, then   c(E) = c(F)
    ("Homeomorphic" is the precise term for topologically "like".)

Using the above 3 properties as axioms, it's not difficult to show by induction that, if it's defined at all, the  c  of n-dimensional space can only be  (-1)n (HINT:  A plane divides space into 3 disjoint parts; itself and 2 others...)

  • c (point) = 1
  • c (entire straight line, or open segment) = -1
  • c (plane or open disc) = 1
  • c (space or open ball) = -1
  • c (space with n-dimensions) = (-1)n
  • c (surface of a sphere) = 2
  • c (surface of an infinite cylinder) = 0
  • c (surface of torus) = 0
  • c (circle, or semi-open segment) = 0
  • etc.

Now, back to our problem:  Why is the Descartes-Euler formula valid to begin with?  Well, that's because the c of a sphere's surface is 2 and it's "made from" disjoint faces, edges and vertices, each respectively with a c of 1, -1 and 1.

In the "natural" breakdown of your cylinder (whose c is indeed 2), you have no vertices, two ordinary faces (whose c is 1) and one face whose c is 0 (the lateral face), whereas the c of both edges is 0. The total count does match.

Note (2000-11-19) :   The orthodox definition of the Euler-Poincaré characteristic does not use the above 3 fundamental properties as "axioms" but instead is closer to the historical origins of the concept (generalized polyhedral surfaces).  It would seem natural to extend the definition of c to as many objects as the axioms would allow.  This question does not seem to have been tackled by anyone yet... 
    Consider, for example, the union A of all the intervals [2n,2n+1[ from an even integer (included) to the next integer (excluded).  The union of two disjoint sets homeomorphic to A can be arranged to be either the whole number line or another set homeomorphic to A.  So, if c(A) was defined to be x, we would simultaneously have x = x+x and -1 = x+x.  Thus, x cannot possibly be any ordinary number, and the latter equation says x is nothing like a signed infinity either  [as (+¥)+(+¥) ¹ -1]. At best, x could be defined as an unsigned infinity (¥) like the "infinite circle" at the horizon of the complex plane (¥+¥ is undetermined).  This could be a hint that a proper extension of c would have complex values...

(2003-11-27)   Generalized Euler Characteristic
Extending the Euler characteristic (1752) to complex values.

 Come back later, we're
 still working on this one...

Just about 3 years after posting the previous article at its original location, we resumed our reflection about an extended Euler characteristic.  The hunch about complex values turned out to be decisive, based on our previous observation that the c of the set A described in the footnote could only be an unsigned infinity...

In the original version of the footnote, we shyly called this a "lame" hint that extended chi-values could be complex.  We've now edited this out!

The set A was clearly a failed attempt at building something with a c of  ½.  [As I recall, finding out it could only be an unsigned infinity was disappointing...]  With hindsight, it's clear that there's a more compelling approach, based on another well-known property of c concerning cartesian products, which is worth preserving in any interesting extension of c

c ( E ´ F )   =   c(E)  c(F)

Using the 3 "axioms" of the previous article [and the value (-1)n which they impose for the c of ordinary n-dimensional Euclidean space] this relation can be easily established by [structural] induction for all "polyhedral" sets.  (Such sets, which are the usual domain of definition of c, consist of finite unions of disjoint components, each homeomorphic to some n-dimensional Euclidean space, which are called its vertices, edges, faces, cells...)  Therefore, the above relation does not contradict our three axioms and may be use as a fourth axiom in a larger scope of more general sets, which remains to be defined...

As we expect complex numbers to be involved, we're also expecting an arbitrary choice between i and -i, probably linked to the chirality of sets so that the chi of a set and of its "mirror image" are complex conjugates of each other.  We are thus led to assume that c is only preserved by homeomorphisms that conserve chirality and could restate the third axiom (C) accordingly, in terms of those homeomorphism which preserve the orientation of an immersing space.

For an homeomorphism which does not preserve such an orientation, it may be possible to find a larger space in which the orientation is preserved whose restriction to a smaller space violates orientation  (a two-dimensional symmetry about a line is a restriction to the plane of a three-dimensional rotation about that line).  This is a clue that an intrinsically  chiral  topological space can't be immersed in a space of finitely many "dimensions".

Let's try to build a set E whose cartesian square E´E has a c of -1...  We would then expect the cartesian product of E and its mirror image to have a c of +1 and this may guide the search...

Consider a Hilbert space with the countable basis denoted |0>, |1>, |2>, |3>, etc.  It is homeomorphic to its own cartesian square  (HINT:  Use the even coordinates of a given ket to form a first ket and the odd ones to form a second ket.)

 Come back later, we're
 still working on this one...

Wikipedia :   Complex Measure

(2007-10-31)   The Real Projective Plane  &  Boy's Surface (1901)
Werner Boy found a 3D  immersion  of the  real projective plane.

The set whose elements are straight lines going through the origin in three-dimensional Euclidean space is known as the  real projective plane.

David Hilbert (1862-1943) did not think the  real projective plane  could be  immersed  as an ordinary surface in 3-dimensional Euclidean space, but he couldn't prove it was impossible.  So, he assigned the task to one of his graduate students,  Werner Boy,  who earned his Ph.D.  in 1901  by finding such an immersion, now called  Boy's surface.

Boy's surface  has Euler characteristic  c = 1.  It can be represented as a single-sided surface with a vertical axis of ternary symmetry.  On that axis is a single pole  P  which looks like an ordinary point from the top but appears from the bottom as the triple point  T  where three  seams  meet  (each such seam is locally equivalent to three flat surfaces sharing an edge, two of those can be smoothly aligned to allow a vantage point where all seams are hidden behind a smooth part of the surface).  Other representations do not break at all any fundamental ternary symmetry.

Werner Boy (1879-1914)  worked several years as a teacher in a Gymnasium of Krefeld  (19 km NW. of Düsseldorf)  before returning to his birth town of Barmen  (now Wuppertal, 28 km E. of Düsseldorf).  He died in the first weeks of World War I  (September 6, 1914).

Wikipedia :   Boy's surface   |   Werner Boy (1879-1914)

(2017-07-25)   Classification Theorem for connected  Closed Surfaces
closed surface  is a compact 2-D manifold without boundary.

A (topological) surface is simply a two-dimensional manifold.  Unfortunately, the traditional locution  closed surface  doesn't necessarily apply to a surface which is merely closed; it has to be compact and  borderless  as well  (i.e., its boundary must be empty).

Connected Sum of two Surfaces :

The connected-sum of two surfaces is obtained by removing an open disk from one and a closed disk from the other and  gluing  the two remaining component so that the two deleted disk share the same border.

 Come back later, we're
 still working on this one...

Classification of closed surfaces   |   Cross-cap

(2006-08-26)   Hadwiger's Theorem  (1957)
About the additive continuous functions of d-dimensional  rigid  bodies.

All the finite unions of convex sets of points in d-dimensional  Euclidean space  form what's called  the d-dimensional  convex ring  (a set of points is convex if it contains all the straight segments whose extremities are in it).

A map is said to be  conditionally continuous  when the images of the convex approximations to a convex body always converges to the image of the body.  The blog of Dan Piponi  may serve as a nice introduction to  Hadwiger's Theorem,  a celebrated theorem of integral geometry published in 1957 by the Swiss mathematician  Hugo Hadwiger  (1908-1981).  Namely:

If it is invariant under translations and rotations in d-dimensional euclidean space, any  finitely additive function  which maps finite unions of convex bodies to  real  numbers must be a linear combination of  d+1  uniquely defined  "n-dimensional content" functions  (where the index n goes from 0 to d).

Hadwiger's "0-dimensional content" function is proportional to the  Euler characteristic  (the chi-function  c )  discussed above.  In 3 dimensions, the "n-dimensional contents" for n=1,2,3 are respectively proportional to the body's mean curvature, its surface or its volume.

 Come back later, we're
 still working on this one...

Hugo Hadwiger  (1908-1981)

(2006-04-24)   Oriented Topological Spaces
A topological definition of orientation applies to  some  spaces.

 Come back later, we're
 still working on this one...

(2007-11-04)   Winding number
Counterclockwise turns of a planar curve around an  outside  point.

Consider a  continuous  oriented planar curve which doesn't go through point  O.

g :   [ a, b ]   ®   R2 -{O}  =  C*

As the point  M = g(t)  moves positively along that curve  (which need not be differentiable)  an observer at the origin  O  may record  unambiguously  the variations of the angle which  OM  forms with some fixed direction  The usual ambiguity modulo 2p does not apply because we are considering a  continuous  variation in an angular difference which starts at zero.

The total change in that angle, expressed in  turns  (the number of radians divided by 2p)  is called the  winding number   W ( g, O )   of  g  around point  O.

If the curve is closed, that winding number is an integer.  For example, it's +1 for any counterclockwise circle going around the origin.  It's -1 if such a circle is oriented clockwise.  It's  0  if the circle does not go around the origin.

The  winding number  around the origin is invariant under reparametrization of the curve and also under homotopy within the  punctured  plane  C*.  This is illustrated by the following popular theorem:

"Dog on a Leash"  Lemma :

If a man and a dog walk respectively around closed curves  g0  and  g1  so that the "leash" segment  [ g0(t), g1(t) ]  never touches the "hydrant"  O,  then:

W ( g0 , O )   =   W ( g1 , O )

This is a consequence of the invariance of the winding number by homotopy, since the following curve is a valid homotopic interpolation within the  punctured  plane  (since  g(t)  is never on the "hydrant", because it's a point of the "leash").

     g(t,s)   =   (1-s) g0(t)  +  s g1(t)    QED

In particular, the lemma applies whenever the distance from the hydrant to the man (or to the dog) is  less than  the length of the leash  (such an inequality ensures that the hydrant cannot be between the man and the dog).

Fundamental Theorem of Algebra :

Any complex polynomial  P  of degree  n > 0  has at least one complex root.

Proof :   WLG, assume that  P  is a  monic  polynomial of leading term  xn.  We aim to apply the above dog on a leash lemma.

For some positive number  r,  let's consider the following closed curves  (as the parameter  t  goes from  0  to  2p).

g0(t)   =   ( r e it ) n
g1(t)   =   P ( r e it )

The winding number of  g0  around the origin is clearly  n  because the argument of  g0(t)  increases by  2pn  when  t  increases continuously by  2p.

Now,  |g1(t)-g0(t)|  is bounded by a fixed polynomial in  r  of degree  n-1.  For a large enough  r,  that's less than |g0(t)| = r n.  Therefore, by the dog-leash lemma,  the winding number of  g1  around the origin is  n  (the same as  g0 ).

If  P  didn't have any zeroes, then   g(t,s)  =  P (  r (1-s) e it  )   would be a valid  homotopic interpolation  (within the  punctured  complex plane)  shrinking  g1  down to a pointlike curve located at  P(0).  This would make the winding number of  g1  equal to  0  instead of  n.  Therefore,  P  must have at least one zero.  QED

(2007-11-11)   Fixed-Point Theorems
For some sets, all continuous mappings have a fixed point.

fixed-point  of a mapping  f  is a point  x  such that  f (x) = x.  An idiomatic way to state the same thing is to say that  f  fixes  x.

The archetypal fixed-point theorem is  Brouwer's fixed-point theorem,  which says that any continuous function from  E  to itself must have a fixed point when  E  is homeomorphic to a closed ball of an n-dimensional Euclidean space.

Compact convex (Schauder)

Locally convex (Tychonoff)

 Come back later, we're
 still working on this one...

In a general topological space, it may be convenient to turn the  fixed-point theorem  into a definition of a specific class of sets.  Let's just call  Brouwerian  a set  E  for which all continuous functions from  E  to itself have at least one fixed point.  The above fixed-point theorems can be expressed by stating that the following sets are  Brouwerian :

  • A set homeomorphic to an n-dimensional closed ball.  (Brouwer)
  • A convex compact subset of a normed vector space.  (Schauder)
  • A locally convex subset of a topological vector space.  (Tychonoff)

Sperner's lemma (1928):  Combinatorial equivalent of Brouwer's fixed-point theorem.
Video :   Fixed points  by  Michael Stevens  (VSauce, 2016-09-28).

(2007-11-04)   Turning Number
The winding number of the nonvanishing  tangent  to a planar curve.

If a point moves in the plane with a continuous nonvanishing velocity, the winding number of the velocity about zero-velocity is well-defined  (it would not be for a curve with singular points where the velocity can vanish, because the winding number is not defined for a curve which goes through the origin).

This number does not depends on the details of the motion, except its orientation.  It's a characteristic of the oriented trajectory called the  turning number.  For a  closed  trajectory, that  turning number  is an integer.

One way to compute this integer for a closed curve is to focus only on those points where the oriented tangent has a given direction  (e.g., due east).  Each such point is assigned a zero value if it's an inflection point, a value +1 if the curve lays to the left of its tangent or a value of -1 if it lays to its right.  The sum of those values is equal to the curve's  turning number.

The Whitney-Graustein theorem states that two closed differentiable curves are homotopic within the plane  if and only if  they have the same turning number.

 Bill Thurston
(2007-10-26)   Eversion of the sphere
A regular homotopy can turn a sphere inside out.

The approach presented in the previous article for closed planar curves can be adapted to three-dimensional oriented smooth surfaces by focusing only on those points where the normal vector is vertical and pointing  upward.  Such points are counted for +1 if the surface does not cross its tangent plane, whereas  proper  saddlepoints are counted as -1.  The sum of all such values is a characteristic of the surface which remains invariant by any homotopy.

A proper saddlepoint is characterized by a second-order variation which is a differential form with two real roots.

In 1957, Steve Smale  (b. 1930, Fields Medal in 1966)  proved that an eversion of the sphere was possible  (this result is so surprising that it's still known as Smale's paradox).  In 1961, Arnold Shapiro came up with the first  practical  eversion of a sphere.  He did not publish it but described it to Bernard Morin.  Morin discussed it with René Thom who exchanged letters about the subject with Tony Phillips.  In 1966, this culminated in a popular article by Philips for  Scientific American  (loosely following Shapiro's original construction).  In 1967, Morin came up with an eversion which was simpler than all previous ones.

Bernard Morin (1931-) has been blind since age 6.  He is a brilliant French mathematician who spent most of his career at the University of Strasbourg.  (François Apéry, son of Roger Apéry, was one of his graduate students.)  He described what's now called the Morin surface as the half-way stage in a superb  eversion  of the sphere.  Morin is also known for having given (in 1978) the first parametrization of Boy's surface  (the 3D immersion of the  real projective plane  found in 1901 by Werner Boy, a student of David Hilbert).

In 1974,  Bill Thurston (1946-2012)  introduced a new sphere eversion based on his  corrugation  method  (illustrated in the video Outside in).

Eversion of a sphere  (Thurston's method)

(2007-10-31)   Classification of Closed Surfaces & Conway's ZIP (1992)
Conway's "Zero Irrelevancy Proof" of the classification theorem (1860).

A connected closed surface is homeomorphic to either

  • A sphere with  n  handles  (orientable, c = 2-2n ).  Such a surface is called an  n-torus:  sphere (n=0), torus (n=1), double-torus (n=2), triple-torus, etc. 
  • A sphere with  n  crosscaps  (nonorientable, c = 2-n ).  Such a surface is called an n-cross surface  (this nomenclature is due to John H. Conway):  The real projective plane is a cross surface, the Klein bottle is a double-cross surface, Dyck's surface is a triple-cross surface.

 Come back later, we're
 still working on this one...

Conway's ZIP  by George K. Francis & Jeffrey R. Weeks

(2007-10-31)   Braid Group  Bn

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 still working on this one...

Emil Artin,
March 3, 1898,
Dec. 20, 1962.
Emil Artin  (1898-1962)

There is a surjective group homomorphism from  Bn  to  the symmetric group  Sn  (the group of all permutations of  n  elements).  The kernel of this group is the  pure braid group on  n  strands  Pn.

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