Timeline for Relative compactness in topological spaces (reference request)
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 26, 2020 at 9:10 | answer | added | Martin Väth | timeline score: 1 | |
| Dec 25, 2020 at 22:50 | comment | added | Jochen Glueck | @MartinVäth: Your comment motivated me to have a look at your book Topological Analysis (2012), where you define relative compactness in Definition 2.31. Within the range of books that I looked at today, this now makes 2:1 for your definition. ;-) (But as mentioned before, I find your definition conceptually quite convincing anyway. So thanks again for your comment.) | |
| Dec 25, 2020 at 16:51 | comment | added | Jochen Glueck | @MartinVäth: Good point. While, in the above question, I'm mostly interested in the Hausdorff case (I just mentioned (i) "for the sake of completenss"), I agree that one might find your definition of relative compactness more natural. I've browsed a bit of literature: Bourbaki use your definition, too (but, as one would expect from Bourbaki, they call it relatively quasi-compact), while Schechter's handbook mentioned in the comments and in the answer uses the definition from my question. Several other books don't mention the term "relatively compact" at all. | |
| Dec 24, 2020 at 12:00 | comment | added | Martin Väth | Concerning the “counterexamples” in (i), I would argue that you use the wrong definition of "relatively compact". In a not necessarily Hausdorff space the natural definition is IMHO the following: $M\subseteq X$ is relatively compact in $X$ if there is a compact $K\subseteq X$ with $M\subseteq K$. With this definition, the “counterexamples” in (i) become empty. | |
| Dec 17, 2020 at 17:16 | comment | added | Renan Mezabarba | I found this book when I was studying about some equivalences of the ultrafilter lemma, back in 2016. Since then it is my favorite book, and even an inspiration for my own book. | |
| Dec 17, 2020 at 17:02 | comment | added | Jochen Glueck | @RenanManeliMezabarba: Thanks for posting the answer! By the way, this Handbook by Schechter is incredible! I've dreamed about such a book for years (but didn't know it really exists - until today). | |
| Dec 17, 2020 at 16:43 | vote | accept | Jochen Glueck | ||
| Dec 17, 2020 at 16:42 | answer | added | Renan Mezabarba | timeline score: 4 | |
| Dec 17, 2020 at 16:42 | comment | added | Renan Mezabarba | Glad to hear that! I'll post it then :) | |
| Dec 17, 2020 at 16:31 | comment | added | Jochen Glueck | @RenanManeliMezabarba: Ha, great! Schechter indeed has the result, namely in Section 17.15 (on page 460). It is even true on regular spaces, not only on completely regular ones. Thanks a lot! If you post this as an answer, I'll of course accept it. | |
| Dec 17, 2020 at 15:36 | comment | added | Renan Mezabarba | Not sure if you will find it there, but I would take a look on the Handbook of Analysis and its Foundations, by Eric Schechter. | |
| Dec 17, 2020 at 12:55 | history | asked | Jochen Glueck | CC BY-SA 4.0 |