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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 30th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2017
   • (edited May 11th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexComing back to this, now that we have _[[closed-projection characterization of compactness]]_: i have cross-linked the entry with _[[tube lemma]]_.

   Coming back to this, now that we have closed-projection characterization of compactness: i have cross-linked the entry with tube lemma.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexAnd I have made explicit in both entries ([here](https://ncatlab.org/nlab/show/closed-projection+characterization+of+compactness#TubeLemma) and [here](https://ncatlab.org/nlab/show/tube+lemma#ProofOfTubeLemmaViaClosedProjectionCharacterization)) the direct proof of the tube lemma via the closed-projection characterization.

   And I have made explicit in both entries (here and here) the direct proof of the tube lemma via the closed-projection characterization.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 11th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOf 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 11th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added what you just said as a remark to the entry, right before and right after the statement and proof [here](https://ncatlab.org/nlab/show/closed-projection+characterization+of+compactness#TubeLemma).

   I have added what you just said as a remark to the entry, right before and right after the statement and proof here.