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(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  each  of its points.

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

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(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.  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.)

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(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 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: adhérence)  Ã 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  boundary)  ¶A  of a set  A  is the intersection of the closure of A with the complement of the interior of A.  ( ¶A is a  closed set,  because it's the intersection of two closed sets.)

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

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

A  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).

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(2007-11-09)   Fine (= strong) or coarse (= weak) topologies.
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.  Instead, each now denotes a particular flavor of topology which may be specified as needed, whenever the corresponding property is relevant.  Here's a lexicon...

The letter  T  is for  Trennung,  a German word meaning "separation" but  (in English at least)  separability is a totally different concept.

The trivial  (or  indiscrete)  cofinite  and  discrete  qualifiers denote specific topologies.  The other qualifiers indicates entire classes of topologies having the properties specified above, which  all  metric spaces  do  satisfy.

History of the separation axioms   |   Topological properties

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(2007-11-09)   Compact Sets
Sets for which any  open cover  includes a  finite  subcover.

The  Heine-Borel Theorem  states that in the Euclidean space  ⁿ,  a set is compact if and only if it's  closed  and  bounded.  More generally, in a 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.

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(2007-11-15)   Completeness
A  metric  concept which is  not  a topological property.

A  metric space  is  complete  when any Cauchy sequence in it converges.

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By definition, a  topological property  is preserved by any homeomorphism.  This is  not  always the case for completeness.  For example,   is complete and it's homeomorphic to the open interval  ]0,1[  which is  not.  (HINT :  A positive sequence that tends to  0  in [0,1]  isn't convergent in  ]0,1[ .)

Thus, completeness is not strictly topological.  Nevertheless, it may be enlightening to attempt topological characterizations of completeness to see how such attempts fail.  For example, let's examine the following statement:

Any decreasing sequence of nonempty closed sets has a nonempty intersection:

{   " i Î ,  Ai ¹ Æ  is closed,   Ai+1 Í Ai   }     Þ     Æ ¹	Ç	Ai
	i Î

This would seem like a good candidate for a topological characterization of completeness until you realize that it's not even true for a noncompact complete space like    in which there are indeed nested collection of nonempty closed sets with an empty intersection.  One example is  A_i = [i,¥[.

For families of  compact  closed sets, the above characterization still fails for metric spaces of infinitely many dimensions (where closed balls are not compact).

All told, a topological space can only be said to be complete when a distance can be defined on it which induces its own topology.  Such a space is called either  topologically complete  or  completely metrizable.  There is simply no easy way to characterize that property...

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(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 metrical concept of completeness.

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(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.

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(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...  A more intuitive definition would be that  a function is continuous if and only if it transforms any connected set into a connected set.

Indeed, this is true, with the definition of a  connected set presented below.

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

• The inverse image of an open set is the intersection of  D  with an open set.
• The inverse image of a closed set is the intersection of  D  with a closed set.
• The direct image of a compact set is a compact set.
• The direct image of a connected set is a connected set.

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(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
	iÎI

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

{ x }   =	Õ	{ pi (x) }
	iÎI

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

If we didn't insist on  U_i  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" toppology 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  f_i  are continuous.

{  f (x) }   =	Õ	{  fi (x) }
	iÎI

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).

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.

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(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  nomempty  parts that are disconnected from each other. 
• A set is said to be  connected  if it's not disconnected.

(Note that a set containing just one point is connected.  So is the empty set.)

In particular, the whole  topological space  E  is  connected  if and only if it doesn't contain any nonempty proper subset which is  both  open and closed.

In a topological space endowed with the discrete topology, there are no connected sets containing more than one point.  On the other hand, with the trivial topology  (the so-called  indiscrete topology)  every set is connected.

Connectedness and Continuity :

As previously advertised, we shall now justify the above definition of continuity by establishing that it matches the more intuitive characterization of a continuous function as something that transforms a connected set into a connected set...

Path Connectedness :

In a  topological space  X,  a  path  from  a  to  b  is 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  when such a path exists  within  Y  for every pair of points  {a,b}  of  Y.

Theorem :   Every path-connected set is connected.

Proof :   Consider two  arbitrary  points  a  and  b  of a path-connected set  Y.  Let  P  be a path from one to the other.  If those two points were respectively in two disjoint open sets  U  and  V  whose union contained  Y,  then such open sets would likewise split  P  and prove it not to be connected.  Since we know that  P  is connected (as a continuous image of the connected set [0,1]) 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.   

Theorem :   Every connected  open  set of a normed space is path-connected.

Proof :   For any point  a  of a nonempty open set  U,  we may consider the set  V  of the points  b  of  U  for which there is a path from  a  to  b.  The set  V  is open because balls in a normed space are path-connected  (HINT:  any point c of a ball centered on b and contained in U is contained in V, because there's a path from a to c which goes through b).  Similarly, the set  W  consisting of the points  z  of  U  for which there is no path from  a  to  z  is also open.  So, U is the union of two disjoint open sets V and W.  If U is connected, this is only possible if W is empty  (since V is nonempty because it contains  a).  Therefore,  U = V, which is to say that there is a path from  a  to any other point of  U.   

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

A  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.

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(2007-10-31)   Homotopy  &  homotopic functions

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.

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(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

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

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

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(2007-10-31)   Homology  &  Cohomology

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(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...

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 equal to (-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...

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(2003-11-27)   Generalized Euler Characteristic
A natural extension of the Euler characteristic, with complex values.

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.)

Wikipedia :   Complex Measure

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(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 a young graduate student of his, Werner Boy (1879-1914).  Against all expectations, Boy made a name for himself by finding such an immersion in 1901.  The Euler characteristic  (c)  of  Boy's Surface  is 1.

Werner Boy 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).

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(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.

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

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(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 ]   ®   ² -{O}  =  *

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  *.  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)   

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 :   Without loss of generality, we assume that  P  is a  monic  polynomial of leading term  xⁿ.  We aim to apply the above dog on a leash theorem.

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 on a 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. 

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

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)

In a general topological space, it may be convenient to turn the  fixed-point theorem  into a definition of 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 restated:

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

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(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.

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(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 in any homotopic transformation.

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.

Amazingly, Bernard Morin (1931-) has been blind since age 6.  Morin 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 (b. 1946, Fields Medal in 1982) introduced a new sphere eversion based on his  corrugation  method  (illustrated in the video Outside in).

Eversion of a sphere  (Thurston's method)

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(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.

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

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(2007-10-31)   Braid Group  B_n

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.