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Martin Sleziak
Martin Sleziak
5:29 AM
72
Q: What's going on with "compact implies sequentially compact"?

mattI've seen both counterexamples and proofs to "compact implies sequentially compact", and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick googling and wikipedia checks will turn up examples floating around; they tend to be variants of $[0...

general-topology compactness
Where can I find a good proof for X being first countable of this result? A book would be the best scenario. This result is going to be used in a journal of Applied Mathematics and I can't assume the reader has such background in Topology, since some of my readers might have graduated in other areas instead of Mathematics. Any help with it would be absolutely grateful. — Renan Willian Prado 36 mins ago
The things which are worth trying, are checking some standard references for general topology:
26
Q: Reference for general-topology

newbieThough there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level. I am wondering if there is a book or a set of books on topology like...

general-topology reference-request big-list book-recommendation reference-works
And maybe also checking the top results in Google, Google Books, Google Scholar.
One of classical text is Engelking's General Topology. I will add a link to my notes: thales.doa.fmph.uniba.sk/sleziak/texty/rozne/engel/engel.pdf (I think I made it based on the older edition.)
> Theorem 3.10.30. Every sequentially compact space is countably compact.
> The reverse implication does not hold; there exist even compact spaces which are not sequentially compact -- by virtue of Corollary 3.6.15, the Čech-Stone compactification $\beta\mathbb N$ issuch a space (cf. Example 3.10.38). We have however
> Theorem 3.10.31. Sequential compactness and countable compactness are equivalent in the class of sequential spaces and, in particular, in the class of first-countable spaces.
One could also check Willard's General Topology, where you can find this result in Problem 17.G.3..
> 1. Not every compact space is sequentially compact. [Consider an uncountable product of copies of $\mathbf I$.]
> 2. Every sequentially compact space is countably compact, but not every sequentially compact space is compact. Hence, together with part 1, sequential compactness is neither stronger nor weaker than compactness; just different. [Use $\mathbf\Omega_0$.]
> 3. A first-countable space is sequentially compact iff it is countably compact. (Thus, for metric spaces, sequential compactness is equivalent to compactness, by 17F.6.)
> 4. A second-countable $T_1$-space is sequentially compact iff it is compact.
@RenanWillianPrado I have mentioned some references in the General Topology chatroom. (Maybe somebody will notice those messages and add some further references.) — Martin Sleziak 33 secs ago
 
Martin Sleziak
Martin Sleziak
6:09 AM
And I am sure that there are also several other posts on Mathematics closely related to this. For example, looking through the posts tagged compactness+first-countable I found the question A first countable, countably compact space is sequentially compact with an answer by Henno Brandsma:
4
A: A first countable, countably compact space is sequentially compact

Henno BrandsmaTheorem: $X$ is countably compact, then $X$ is strongly limit compact: every countably infinite subset $A$ has an $\omega$-limit point, i.e. a point $x$ such that for every neighbourhood $U$ of $x$ we have $U \cap A$ is infinite. Proof: suppose not, then every $x \in X$ has a neighbourhood $O_x$...

Some related questions can be found also among linked questions to some of the posts mentioned above: math.stackexchange.com/questions/linked/44907 math.stackexchange.com/questions/linked/152447
 
 
13 hours later…
Martin Sleziak
Martin Sleziak
7:22 PM
It's all good references for sure! I can't express my gratefulness for the Theorem 3.10.31 in Engelking's General Topology. — Renan Willian Prado 32 mins ago
 

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