From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Beginner question on differential geometry
Date: 18 Nov 1997 21:28:57 GMT

In article <3468A443.600639BF@dnai.com>,
Jeffrey B. Rubin <jeffr@dnai.com> wrote:
>In the very beginning of Spivak's Comprehensive Intro to Differential
>Geometry, he quotes the Invariance of Domain theorem from
>Algebraic Topology and then says that it is trivial to show that it
>implies that the neighborhood of a point that is homeomorphic
>to R^n in his definition of a topological manifold must be open.
>Can somebody tell me what this trivial line of thought is? 

An n-dimensional manifold is defined to be a Hausdorff topological space  M  
in which every point  x  has a neighborhood  U  homeomorphic to  R^n. The 
problem at hand is to show this  U  is an _open_ subset of  M.

Pick any  y  in  U; we must show there is an open set of  M  containing  y  and
contained in  U. Now, by assumption  y  itself has a neighborhood  V
homeomorphic to  R^n. To call it a neighborhood of  y  is to say that its
interior  W  (the maximal open subset of  M  contained in  V) contains  y.

By definition of the subspace topology, W'=  W intersect U  is an open subset
of  U . But then under a homeomorphism f: U -> R^n, W'  maps to an open subset
W''  of  R^n. Now turn this around:  f^(-1) is a homeomorphism from
W'' to  W', which you can follow with the (continuous, 1-to-1) inclusions
W' -> U and U -> M. Of course, the image is contained in   W  and hence in
V, so we can follow with another homeomorphism  g: V -> R^n.

Thus we have a continuous, 1-to-1 map from an open subset  W'' of  R^n  to
R^n. Using the Invariance of Domain, its image is open in  R^n. Its inverse
image under  g, namely  W', is then open in  V, and hence in  W (subspace
topology again). But  W  is actually  open in  M  by construction, that is,
W intersect U  is open in  M. This is the open set in  M  containing  y  which
we sought.

Invariance of Domain is one of those theorems which are a little hard to
appreciate because at first blush they appear to be telling you something
which is trivial; but after you raise objections of the sort you did,
you can appreciate that there's really something nontrivial there (which 
is why it takes such tools to prove!)

dave