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