Timeline for Why does normality imply that a countable base $B$ contains at least one set $U$ whose closure is a subset of another set $V \in B$?
Current License: CC BY-SA 4.0
12 events
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| Jul 29, 2022 at 16:48 | comment | added | rmoron | Sorry, yes, I'm not particularly familiar with this syntax, yet. Thanks. | |
| Jul 28, 2022 at 23:54 | comment | added | LSpice |
Was there meant to be a link where you say "PDF available at link"? The DOI is <doi.org/10.1007/978-3-662-03004-2>, if that's what you wanted. (By the way, I notice you rightly edited out the \[ \], since the semantically correct way of doing it is not supported by MathJax; but, if you want a displayed environment, then the old-style $$ $$ works just fine.)
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| Jul 28, 2022 at 22:21 | comment | added | rmoron | With that in mind I've removed "proper" from the question. I think this can be considered answered now. | |
| Jul 28, 2022 at 22:20 | history | edited | rmoron | CC BY-SA 4.0 |
edited title
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| Jul 28, 2022 at 22:17 | comment | added | rmoron | @AlessandroCodenotti I think $V=U$ is the right solution for the countable case. I had wondered about that and checked the notational conventions as addressed in the book as far as proper subsets, but I wasn't confident of my reasoning, and the character limits prevented me from including this point in the question. | |
| Jul 28, 2022 at 21:32 | comment | added | Alessandro Codenotti | Also I think this can be shown by an argument similar to the one used to show that a regular space has a local basis of closed sets at every point with a couple extra steps | |
| Jul 28, 2022 at 21:24 | comment | added | Apollo | @alessandro-codenotti - That makes sense, though I would not call it a "proper" subset. (It may be that the text does not require that and was just assumed by the OP, forgetting about the possibility of clopen sets.) | |
| Jul 28, 2022 at 21:17 | comment | added | Alessandro Codenotti | @Apollo I suppose the inclusion is not meant to be strict, so that if $U$ is clopen as in your example you can take $V=U$ | |
| Jul 28, 2022 at 20:46 | comment | added | Apollo | There must be more conditions than you've stated here. For example, the discrete space on a countable set X has a countable basis (consisting of the (clopen) singleton sets), but no set A in that basis is a proper subset of any set B in the basis. Note that the discrete topology is Hausdorff and normal. | |
| Jul 28, 2022 at 20:10 | history | edited | Michael Albanese | CC BY-SA 4.0 |
deleted 7 characters in body
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| S Jul 28, 2022 at 18:06 | review | First questions | |||
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| S Jul 28, 2022 at 18:06 | history | asked | rmoron | CC BY-SA 4.0 |