If a T1 topological space is closed under countable intersections, does this necessarily make the topology discrete? It is easy to construct a counterexample if the topological space is not assumed to be T1, and it is also easy to see that the claim is true in a metrizable topological space. If the claim is false under the assumption that the space is T1, then what is the weakest condition required to ensure the topology is discrete?

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    $\begingroup$ Consider a cocountable topology for a counterexample. A condition under which it is true is first countability, and I don't think you can do much better than that. $\endgroup$ – Wojowu Jan 28 at 16:38
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    $\begingroup$ Keyword: P-spaces. $\endgroup$ – Ramiro de la Vega Jan 28 at 18:22
  • $\begingroup$ Also a compact Hausdorff space in which the topology is closed under countable intersection, is finite (this is mentioned in Rudin, Duke 1956). Indeed, the assumption implies that all countable subsets are closed, while every infinite Hausdorff compact space has an infinite countable discrete subset, necessarily non-closed. $\endgroup$ – YCor Jan 28 at 19:01
  • $\begingroup$ A Hausdorff non-discrete topological space in the topology is stable under countable intersections is as follows: consider a set $X\sqcup\{\infty\}$ with $X$ uncountable, and say that a subset $U$ is open if either $\infty\notin U$, or if $U$ has countable complement. Clearly this satisfies the required conditions. (Actually, all its countable subsets are closed and discrete.) $\endgroup$ – YCor Jan 28 at 19:05
  • $\begingroup$ Every connected regular space where the countable intersection of open sets is still open is still open is a single point. $\endgroup$ – Joseph Van Name Jan 28 at 22:18

Here I am gathering information in the comments along with some information of my own to form an answer.

A $P$-space is a regular space where every countable intersection of open sets is open. There are many examples of $P$-spaces which are not discrete. For example, the co-countable topology is a topological space $(X,\mathcal{T})$ where $X$ is an uncountable set and $\mathcal{T}$ consists of all subsets $R\subseteq X$ where $|R^{c}|\leq\aleph_{0}$ or $R=\emptyset$. The co-countable topology is a $T_{1}$-space where the intersection of countably many open sets is still open.

An Alexandroff space is a space where the intersection of arbitrarily many open sets is open. The only $T_{1}$-Alexandroff spaces are the discrete spaces.

Here are a few examples of benign topological properties for which the only $P$-spaces which satisfy those properties are the discrete spaces. From this point on, let us limit our scope to only the regular spaces.

Connectedness: Every $P$-space is zero-dimensional (a zero-dimensional space is a space with a basis of clopen sets). Therefore, any connected or locally connected regular $P$-space consists of a single point. Furthermore, any regular space $X$ with a connected subset consisting of more than one element cannot be a $P$-space.

Disconnectedness: A topological space $X$ is said to be extremally disconnected if for each open set $U$, the closure $\overline{U}$ is still open. A cardinal $\lambda$ is said to be below the first uncountable measurable cardinal if there does not exist a non-principal ultrafilter $M$ on $\lambda$ such that the intersection of countably many elements of $M$ still belongs to $M$. Every uncountably measurable cardinal is very large and it is consistent with ZFC that there are no measurable cardinals. Every extremally disconnected $P$-space of non-measurable cardinality must be discrete, but there are extremally disconneced $P$-spaces of measurable cardinality such as the one in this answer (this space happens to be the poset used for Prikry forcing).

Compactness: Every compact $P$-space is discrete and finite. In fact, in every $P$-space, every countable subspace must be closed. However, every infinite compact space must have infinite discrete non-closed subspace, so infinite compact spaces cannot be $P$-spaces at the same time. If $X$ is a $P$-space which is pseudocompact, countable compact, or locally compact, then $X$ must be discrete.

Countability: Suppose that $L$ is one of the following properties: metrizable spaces, second countable, first countable, points are $G_{\delta}$, sequential spaces. Then the only $P$-spaces that satisfy property $L$ are the discrete spaces.

Non-examples: There are many topological properties for which there are many non-discrete $P$-spaces which satisfy those properties including the following (somewhat redundant) list: completely regular, normal, paracompact, Lindelof, realcompact, scattered, the Baire property and its generalizations, the negation of the Baire property.


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