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Hi¡ I have some troubles with the next problem.

Let $(X,\tau)$ be a topological space. Prove that $X$ is $T_1$ if and only if there exists a family $U$ of open sets such that $\bigcap U=\Delta$ where $\Delta=\left\{(x,x): x\in X\right\}\subseteq X\times X$ (the diagonal)

My attempt.

$\Rightarrow)$ We know that $X\times X$ is $T_1$ because $X$ is $T_1$. Moreover, we have the next theorem for $T_1$ spaces

Theorem Let $X$ be a topological space. The next conditions are equivalent.

1) $X$ is $T_1$

2) For all $B\subseteq X$, $B=\bigcap \left\{ U: B\subseteq U, U\in\tau\right\}$

3) For all $x\in X$, $\left\{x\right\}=\bigcap\left\{U : U\in\tau, x\in U\right\}$

Then, the implication follows from 3).

But, what can I do for $\Leftarrow)$? I have tried to prove that $\left\{x\right\}$ is closed, but I have failed. My best attempt was consider two distinct points $x$ and $y$. Clearly, $(x,y)\notin\Delta$, then, $(x,y)\notin\bigcap U$, so, there exist some basic open set such that $(x,y)\notin A\times B$. Then, my idea was use the projection $\Pi_X$, but, from here, I'm so confused.

Thanks in advance.

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marked as duplicate by Lord Shark the Unknown, Niels J. Diepeveen, Henno Brandsma general-topology Jun 4 '17 at 13:50

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  • $\begingroup$ What implication? Left to right or right to left? $\endgroup$ – William Elliot Jun 4 '17 at 10:20
  • $\begingroup$ Right to left, clearly. $\endgroup$ – Carlos Jiménez Jun 4 '17 at 10:22
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There is some open U containing the diagonal D, with (x,y) not in D.
U is a union of open base sets U_j, none of which contains (x,y).
As (x,x) is in D, there is some U_j with (x,x) in U_j.
x is in open pi_2(U_j) and y is not.

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