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Do spaces where all singletons are closed have a name? I know for example that $\mathbb R$ is one of these spaces since the complement of a singleton $\{x\}$ is $(-\infty,x)\cup (x,\infty)$ which is open. I know also that a space where all singletons are open is a discrete space since if every singleton is open in $X$ then this would imply that every subset of $X$ is open in $X$. Thank you for your help!!

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up vote 5 down vote accepted

They are called $T_1$-spaces.

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I see in the wikipedia page that the examples of non $T_1$ spaces are not "familiar" spaces, does this mean that being $T_1$ is a rather weak condition that all familiar spaces have ? –  palio Aug 1 at 8:56
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@palio It depends on your definition of familiar, but yes, I'd say so. In terms of nets, being $T_1$ means that a constant net $(x)_{i\in I}$ only converges to $x$, but not to any other point. Thinking of "familiar", that's a property I wanted to have. –  martini Aug 1 at 9:00
    
Given that any union of finitely many closed sets is closed, what does this imply on subsets of $T_1$ spaces ? It means that any finite subset of a $T_1$ space is a closed subset ? do we have a stronger statement? –  palio Aug 1 at 9:09
    
Stronger in what sense? In general there need not be more closed sets, see your favourite infinite set with the cofinite topology (mentioned in the above wikipedia article as a $T_1$-but-not-$T_2$-space), –  martini Aug 1 at 9:10
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As I wrote, in general, more than "finite" cannot be achieved. –  martini Aug 1 at 9:26

To sum up: From the comments above it follows, that every topological space X with topology $\tau$ is $T_1$ if and only if it contains the cofinite topology on X

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