Re: TOPOLOGY: Compactness and T2 - fix

"quasi" <quasi@xxxxxxxx> escribió en el mensaje
news:b048e3der04oon4hmcrbsqm7p3df7vu9ul@xxxxxxxxxx
On Sun, 9 Sep 2007 16:32:20 +0200, "Francisco"
<francisco.bravo.diez@xxxxxxxxxxxxxxx> wrote:

Let (X,T) be a compact, T2 (Hausdorff) topological space,
having a single non-isolated point x_0. Let U be a subset
of X with x_0 in U.

Prove that U is an open <=> X-U is finite.

Hints for the => direction:

(1) Suppose U is open.

(2) Let x be an element of X-U. Then {x} is open. Explain.

(3) Cover X using U as well as all the singletons from X-U.

Does this implication need the T2 premise?

As far as I can see, the => implication doesn't need T2. In fact, even
the <= implication doesn't need it -- T1 would suffice.

Thanks for your help, but I can't figure out why X\U finite, X compact,
X T1 => X\U closed. Would you please elaborate on that? (sorry
about my ignorance).

Hints for the <= direction:

(1) Suppose X-U is finite.

(2) Then X-U is closed. Explain.

For the <= direction, compactness is irrelevant.

For which type of space is a singleton always closed?


For T2. OK, I see it pretty clear now. Many thanks for the illustration ;)


Francisco


.

Relevant Pages

  • Re: TOPOLOGY: Compactness and T2 - fix
    ... Hints for the => direction: ... Cover X using U as well as all the singletons from X-U. ... Does this implication need the T2 premise? ... Thanks for your help, but I can't figure out why X\U finite, X compact, ...
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  • Re: TOPOLOGY: Compactness and T2 - fix
    ... Hints for the => direction: ... Cover X using U as well as all the singletons from X-U. ... Does this implication need the T2 premise? ... about my ignorance). ...
    (sci.math)
  • Re: TOPOLOGY: Compactness and T2 - fix
    ... Let be a compact, T2 topological space, ... Cover X using U as well as all the singletons from X-U. ... Does this implication need the T2 premise? ...
    (sci.math)
  • Re: TOPOLOGY: Compactness and T2 - fix
    ... Cover X using U as well as all the singletons from X-U. ... Does this implication need the T2 premise? ... Thanks for your help, but I can't figure out why X\U finite, X compact, ... a space is T1 iff all singleton sets are closed. ...
    (sci.math)
  • Re: TOPOLOGY: Compactness and T2 - fix
    ... Cover X using U as well as all the singletons from X-U. ... Does this implication need the T2 premise? ... Thanks for your help, but I can't figure out why X\U finite, X compact, ... a space is T1 iff all singleton sets are closed. ...
    (sci.math)