Many proofs of the fact that the product of compact spaces use the tube lemma, and therefore start somewhat like this: Let $X$ and $Y$ be compact spaces. Then for each $x\in X$, $\{x\}\times Y$ can be covered by finitely many open sets. Why is this true? I understand intuitively but am having a hard time rigorously justifying this.
Step in proof that product of compact spaces is compact
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$\begingroup$
general-topology
compactness
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1Let O be open in X and U be open in Y. Then O X U is open in X x Y. Now Y can be covered by finitely many sets W because it is compact. Let x in O an open set in X. Then {x} X Y can be covered by a finitely many O X W. – 2017-02-20
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0For proof: https://en.m.wikipedia.org/wiki/Tychonoff's_theorem – 2017-02-20
2 Answers
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Consider how the topology of $\{x\}\times Y$ is defined, and you will see that $f(p)=(x,p)$, for $p\in Y,$ is a homeomorphism from $Y$ to $\{x\}\times Y.$ Compactness is a topological property, that is, it is preserved by homeomorphism. And $Y$ is compact, so $\{x\}\times Y$ is compact.
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The subspace $\{x\}\times Y$ is homeomorphic to $Y$, which is compact by assumption. Compactness is a topological property, meaning if $X\cong Y$ and $X$ is compact, then $Y$ is compact as well.