# Picturing Urysohn's Metrization Theorem and Urysohn's lemma?

In my Topological course we have this lemma.

[Urysohn's lemma] Suppose that $X$ is a topological space. Then $X$ is normal if and only if, for each pair of disjoint closed subsets $A$ and $B$, there is a continuous function $f:X \rightarrow [0,1]$ such that $f(A)=\{0\}$ AND $f(B)=\{1\}$.

Is there an easy way to visualize this? Also, is there an easy way to understand the proof of this.

The point is this is used to prove. Urysohn's metrization theorem, that is suppose $X$ is a Hausdorff, normal space which has a countable basis then $X$ is metrizable.

But, yeah I need to know the proofs of these for the exam. But, I can't picture them. Don't want to just memorize through, I could do that easily as 5 weeks is when the exam is on. If I look at the proof everyday for 5 minutes, pretty sure I can rote learn it. But, surely there is an easy way to picture this.

If you know a good book on this please say as well. The books I've looked in Munkres and Armstrong, however not that helpful.

One of the thing I hate about Analysis, is that the proofs aren't visual. Heavy definition type proofs with weird tricks. Hard to see what going on.

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I do think I recall that the (German) book by Jähnisch, "Topologie" contained many visualizations, even for theorems for which one would not believe there are any. Maybe you have a copy at your local library -- you don't have to know German, just search for the names of the theorems. It's been a while, though, so don't hold your breath. My memory might fool me. – user20266 Dec 14 '11 at 17:43
"The books I've looked in Munkres and Armstrong, however not that helpful." Weird, since I was going to suggest Munkres. I actually cannot imagine how one could do any better than what's in there... – Sam Dec 14 '11 at 18:11
wikipedia article has a picture illustrating the proof of Urysohn's lemma. Something similar can probably found in many books and lecture notes too, quick google search led me to Patty: Foundations of topology, p.187. (The first google books result which had a picture, I don't know anything more about this book.) – Martin Sleziak Dec 14 '11 at 18:12
@Martin That looks a lot like the picture in Munkres, if I remember correctly. Good find, though! – Dylan Moreland Dec 14 '11 at 23:51
@Dylan: It was not that difficult to find - I simply tried searching for Urysohn lemma at google images and checked the first results at google books. You are right that almost the same picture is in Munkres. (On p.209 in 2nd Edition, it seems that there is no preview available at google books. – Martin Sleziak Dec 15 '11 at 7:03