Talk:Tube lemma

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[edit] Confusion in the proof

I think there is some confusion in the proof: in the definition of $π Y C$ , the $\exists x$ should be deleted: it is the x in the lemma. Second, $π Y$ is used with two different meanings.

I would write something like:

Let $\pi_Y: X \times Y \longrightarrow Y$ be the projection on the second factor. If $(U_{\alpha})_{\alpha \in A}$ is an open cover (in $X \times Y$ ) of $\{x\} \times Y$ , then $(\pi_Y(U_{\alpha}))_{\alpha \in A}$ is an open cover of the compact space $Y$ , from which we can extract a finite open subcover, which corresponds to a finite family $(U_{\alpha})_{\alpha \in B}$ , which covers $\bigcap_{\alpha \in B} \pi_X(U_{\alpha}) \times Y$ . The intersection being a finite intersection of neighborhoods of ${x}$ , the lemma is proved.