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Martin Sleziak

07:19

A: Stone-Čech compactifications and limits of sequences

EDIT: As pointed by Stefan H. in his comment, the solution I have suggested only works if $X$ is normal, since I am using Tietze extension theorem. Perhaps I have overlooked something and I will be blushing, but I will give it a try. (This is my solution, I did not check the books I mentioned...

why is $\{x_n\}\cup\{x\}$ compact in $\beta X$? I can't see this immediately — Westlifer yesterday

@Westlifer We assume that $(x_n)$ converges to $x$. So any open neighborhood $U$ of $x$ contains all but finitely many elements of this sequence. In other words, if you have an open cover of $\{x_n\}\cup\{x\}$ and you choose some $U\ni x$, it only remains to cover finitely many points. — Martin Sleziak 20 hours ago

Q: Stone-Čech compactifications and limits of sequences

I've been working on some old prelims from my university when they used to just be on point-set topology. We don't cover a couple of the topics so I've been teaching myself some of the material, one of the topics no longer covered is Stone-Čech Compactification. Which I have a somewhat tenuous un...