The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom.
My professor asked to prove the following Lemma.
The Tube Lemma: Let $K$ be a compact topological space and $Y$ any topological space. Suppose $p\in Y$ and $U$ an open subset of $K\times Y$ containing the entire slice $K\times \{p\}$. Then we can find an open subset $V$ of $Y$ such that $p\in V$ and $K\times V\subset U$.
This is what I understand:
- A space X is compact provided that every open cover of X has a finite subcover.
- Given that $K$ is compact, $p\in Y$ and $K\times\{p\}\subset U\subset K\times Y$. I seek to prove $\exists V\subset Y:p\in V, K\times V\subset U$.
Here is my rough attempt at the proof:
Let $p\in Y$ and let $U\subset K\times Y$ be open in the product space such that $K\times \{p\}\subset U$. For every $x\in K$, choose open set $A_x\subset K$ and $B_x\subset Y$ such that $(x,p)\in A_x\times B_x\subset U$. Since $K$ is compact, we can select a finite subcover $A_{x_1},...,A_{x_n}$ such that $K=\bigcup_{i=1}^nA_{x_i}$. Let $V=\bigcap_{i=1}^nB_{x_i}$. Then $V$ is a desired neighborhood of $p$. Suppose that $(x,p)\in K\times V$. Then there is an $j$ such that $x\in A_{x_j}$ as the collection $\{A_{x_i}\}_{i=1}^n$ covers $K$. As $p\in V =\bigcap_{i=1}^nB_{x_i}$, we have $(x,p)\in A_j\times B_j\subset U$. This proves $K\times \{p\}\subset K\times V\subset U$.
Am I on the right track? Did I do anything wrong? Is there anything I need to change regarding the proof? Is there any alternative way to proof the Tube lemma that would be much more efficient or detailed?
I sincerely thank you for taking the time to read this question and my attempt at proving it. I greatly appreciate any assistance you may provide.