Abstract Nonsense

Crushing one theorem at a time

Local Properties of Manifolds


In this post we will discuss some of the local properties manifolds attain from n dimensional Euclidean space. It should be intuitive that all of the local attributes should transfer since manifolds “locally look like Euclidean space”. We formalize them in what follows.

We start with a theorem which when put together with the definition of manifolds will give us an amazing revelation.

Theorem: Let \mathfrak{M} be a n-manifold. Then, \mathfrak{M} is locally compact.

Proof: Let x\in\mathfrak{M} be arbitrary. As was proven earlier every manifold has an open base of Euclidean balls, and so let U be the guaranteed Euclidean ball containing x. So, by assumption U\overset{\varphi}{\approx} B for some open ball B\subseteq\mathbb{R}^n. So, by the regularity of \mathbb{R}^n there exists some open ball B'\subseteq B such that \overline{B'}\subseteq B. So, \varphi^{-1}(B')=U' is a neighborhood of x in U. Also,

\overline{U'}=\overline{\varphi^{-1}\left(B'\right)}=\varphi^{-1}\left(\overline{B'}\right)\subseteq\varphi^{-1}\left(B\right)=U

And so, in particular

\varphi\mid_{\overline{U'}}:\overline{U'}\to\varphi\left(\overline{U'}\right)=\overline{B'}

is a homeomorphism and so \overline{U'} is compact in U. Thus, U' is a precompact neighborhood of x in U. But, since compactness in a subspace implies compactness in the ambient space and U is open we see that U' is in fact a precompact neighborhood of x in \mathfrak{M}. The conclusion follows. \blacksquare

The following corollary, to me, was astounding albeit intuitively clear.

Corollary: Since every locally compact Hausdorff space is regular and every second countable regular space is normal we see in particular that every manifold is second countable and normal and thus metrizable by Urysohn’s embedding theorem.

Also, it follows that every manifold which is not compact has an Alexandroff compactification.

We next discuss some issues related to local connectedness/path connectedness.

Theorem: Let \mathfrak{M} be a n-manifold. Then, \mathfrak{M} is locally path connected and consequently locally connected.

Proof: Let x\in \mathfrak{M} be arbitrary and let U be the guaranteed Euclidean ball around it. By assumption U\approx B for some open ball B\in\mathbb{R}^n. But, as was proven earlier every open ball in a normed vector space is convex and thus path connected. Thus, U is path connected and the conclusion follows. \blacksquare

This was just a quick post to prove some results which are true, but need to be proven at least once.

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April 30, 2010 - Posted by | Manifold Theory, Topology | , , ,

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