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.