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I saw tube lemma, and decided to bulk it up.
Many books (such as the famous topology text by Munkres) give proofs which involves multiple subscripts and multiple choices; I’ve written arguments to mostly eliminate that.
Coming back to this, now that we have closed-projection characterization of compactness: i have cross-linked the entry with tube lemma.
Of course the tube lemma was effectively proved within the proof of proposition 1.2. But it’s good to mention explicitly.
More categorically, the tube lemma becomes the statement that for $X$ compact, the map $\forall_\pi: P(Y \times X) \to P(Y)$ takes opens to opens, as mentioned in point 5. of the Variant proofs section.
I have added what you just said as a remark to the entry, right before and right after the statement and proof here.
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