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I'm struggling with grasping the notion of open sets as they apply to topological spaces. For example, if we consider the finite point exclusion topology, $T$, defined by $$T = \{ A \subseteq X \mid p \notin A \ \text{or} \ A = X\}$$any singleton set of the form $\{x \mid x\in X\}$ belongs in $T$, so it is open. However, this set is obviously closed in $X$. How does one distinguish?

What got me thinking about this was a problem that asked what the closed sets in the exclude point topology are. Clearly all singleton sets and their finite unions, including $X$ itself, are closed sets in $T$. But those sets are also open? I don't understand.

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  • $\begingroup$ In a general topological space sets may be open and closed. This does not correspond to our intuitive understanding of open and closed. $\endgroup$ – TheGeekGreek Jun 18 '17 at 16:12
  • $\begingroup$ @TheGeekGreek I apologize if my questions seem rather simple, as I've just started learning the subject. Does this mean that a set can be closed in $X$ while still being "open" in a topology on $T$? $\endgroup$ – user312437 Jun 18 '17 at 16:13
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    $\begingroup$ @JoséCarlosSantos I think I understand. A set $A \subset (X,T)$ is open implies $A\in T$? $\endgroup$ – user312437 Jun 18 '17 at 16:25
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    $\begingroup$ @CodyButler No. A set $A\subset X$ being open is equivalent (by definition) to $A\in T$. $\endgroup$ – José Carlos Santos Jun 18 '17 at 16:27
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    $\begingroup$ @CodyButler That's right! $\endgroup$ – José Carlos Santos Jun 18 '17 at 16:32
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In general a set can be both open and closed. In any topology on $X$ there are at least two examples of such sets, that is $\emptyset$ and $X$ itself. If there are more subsets of the topological space $(X,T)$ that are both open and closed, we say that the topological space $(X,T)$ is disconnected.

But this isn't the case in your question. Note that for the topology $T = \{A \subset X \mid p \not \in A \text{ or } A =X\}$, a subset $A \subset X$ is open if and only if $A$ doesn't contain $p$ or is equal to $X$. A set $B \subset X$ is closed if and only if $B^c$ is open. So $B$ is closed if and only if it contains $p$ or is equal to the empty set. Thus the only sets that are both open and closed are the empty set and $X$ itself.

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  • $\begingroup$ Very clear. Thank you $\endgroup$ – user312437 Jun 18 '17 at 16:33
  • $\begingroup$ You're welcome. $\endgroup$ – Demophilus Jun 18 '17 at 16:34
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Your intuition from the topology of $\Bbb R^n$ and metric spaces doesn't always transfer that well to general topological spaces, which can be quite strange. For example, in an arbitrary topological space, singletons need not be closed. (FYI, a space where every singleton set is closed is called a $T_1$-space, and there are certainly examples of non-$T_1$ spaces.)

Let's look at your example. Recall that by definition, a subset $A$ of a topological space $(X,T)$ is open if $A\in T$. The notion of closed-ness is also completely dependent on $T$: a subset $A$ is closed in $(X,T)$ if its complement is open (that is, $X\setminus A\in T$). Say $x\in X$. If $x\neq p$, the set $\{x\}$ is open: $p\not\in\{x\}$. In this case, $\{x\}$ is not closed: $p\not\in\{x\}$, which implies that $p\in X\setminus\{x\}$, so $X\setminus\{x\}\not\in T$. If $x = p$, $\{p\}$ is closed: $p\in\{p\}$, so that $p\not\in X\setminus\{p\}$, which means $X\setminus\{p\}$ is open. We also see that $\{p\}$ is not open, unless $X = \{p\}$: $p\in\{p\}$, so $\{p\}\not\in T$ unless $\{p\} = X$.

One can also find examples of topologies where singletons may be both open and closed (like the discrete topology on a space), either open or closed (like your example), or neither open nor closed (the generic point is neither open nor closed in the Zariski topology on $\operatorname{Spec}\Bbb Z$).

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  • $\begingroup$ Very detailed and clear. Thank you $\endgroup$ – user312437 Jun 18 '17 at 16:39

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