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 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