Tag Archives: continuous

Why is the Jordan curve theorem difficult?

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So, I decided to rearrange my course material and prove the Jordan curve theorem this week.

The JCT has a reputation for being hard to prove, and it is indeed a non-trivial result. But people don’t always understand why it is non-trivial.

For most curves that one thinks of, the theorem is elementary.  For example, I’ll review in class a very simple proof for polygonal curves, essentially that given in Courant and Hilbert’s classic What is Mathematics?  A similar proof works for differentiable or piecewise-differentiable curves.

Ross and Ross have an article called The Jordan Curve Theorem Is Nontrivial.  It contains some beautiful artwork – representational pen-and-ink drawings made with a single Jordan curve.  Nevertheless, the curves in these artworks are smooth; for them, JCT is elementary.

The real issue is the possibly exotic behavior of topological curves. For instance, the Julia sets of certain complex maps are topological circles, but unlikely to be smooth.  In the topological world, strange things can happen that do not happen in the smooth category.  A famous example is the Lakes of Wada: three connected open sets in the plane which share the same common boundary.  Of course, this boundary is not a Jordan curve.  But who is to say that it might not be!

It is this topological problem which is at the heart of the proof of the full version of JCT.

 

Homework 2, q3 flakiness

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Hi all

There are a couple of points in homework 2 question 3 which seem to have caused some confusion. These are

  • Topologies on function spaces
  • Base point issues

I should have been clearer that you should not worry too much about the first; and as far as the second goes, I made things unnecessarily complicated by the way I worded the question.  My apologies.  (I should add that these are both slightly sophisticated issues; the basic argument for this question can be understood without worrying about either of the points I will bring up below. If none of what I am about to say makes any sense, just ignore it.)

Function space topologies I asked you to prove various loop and path spaces are contractible, and I defined contractibility in terms of the function space \(X^X\). Since loop and path spaces are seldom compact, I have not told you what it means for a path of maps from such a space to itself to be continuous.  So, how to do the question?

The basic answer is that the contraction you produce is given by such a simple formula that it should be continuous under any “reasonable” definition (i.e., “don’t sweat it”).  But I know that this is not satisfactory for some of you.  The appropriate topology for function spaces in this context is usually the compactly generated modification of the compact-open topology.  My British colleague Neil Strickland  has some nice notes about this topology, why it is good, and why it does what you need for function spaces; you can find those here. (An earlier reference is Steenrod’s A Convenient Category of Topological Spaces.)

Base points I screwed up here by using base points in one part of the question but not in another part.  I should have gone “all or nothing”.   Since in the second and third parts of the question I use based loops and paths, the appropriate thing to do would have been to define contractibility in the first part of the question in terms of the path-connectedness of the space of basepoint-preserving maps (rather than all maps) from \(X\) to \(X\).  (Alternatively, I could have kept the first part as it was and worked in the second two parts with unbased loops and paths).

There are examples where “contractibility” and “based contractibility” are not the same thing, though that is impossible for ‘nice’ spaces such as CW-complexes.  See here for some more information about this.

John