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I'm not quite sure where such a concept fits. Suppose $X$ is a topological space. I know then that the diagonal $\Delta=\{(x,x)\ | x\in X\}$, so $\Delta\subseteq X\times X$. What then would a neighborhood of the diagonal be comprised of?

By this I mean, suppose $E\in\mathscr{N}_\Delta$, where $\mathscr{N}_\Delta$ is the filter of all neighborhoods of the diagonal. Is $E\in\mathscr{P}(X)$, or is $E\in\mathscr{P}(X\times X)$? What is the criterion for a subset to be in a neighborhood of $\Delta$? My guess is that $E\in\mathscr{P}(X\times X)$, and if this is correct, we would say

$$(x,x')\in E\quad\text{if and only if...}?$$


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This question is answered in the Wikipedia article on neighborhoods. – Qiaochu Yuan Nov 30 '10 at 23:19
@Qiaochu, thanks, I looked there at first, but missed the article on product spaces. I didn't see it mentioned in the article just on neighborhoods. – yunone Nov 30 '10 at 23:32
up vote 5 down vote accepted

A neighborhood of the diagonal $A$ is a subset of $X\times X$ containing an open set $E$ in $X\times X$ with the product topology such that $\Delta\subset E$.

The set of all neighborhoods of the diagonal is a subset of $\mathcal{P}(X\times X)$.

You might be interested in checking uniform spaces.

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Thank you. It makes sense now. – yunone Nov 30 '10 at 23:25

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