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By a $T_3$ topological space I mean a space in which a point $x$ and a closed set $A$ which does not contain $x$ can be separated by their open neighborhoods, i.e. there are $U,V\in\mathscr{O}$ such that $A\subseteq U$ and $x\in V$ and $U\cap V=\emptyset$.

Could you please point to me some classical example of a space which satisfies the following condition ($\mathscr{O}^+=\mathscr{O}\setminus\{\emptyset\}$ and $\mathrm{Cl}$ is the standard closure operator): $$\forall_{U\in\mathscr{O}^+}\exists_{V\in\mathscr{O}^+}\,\mathrm{Cl}\, V\subseteq U$$ but is not $T_3$?

Rafał Gruszczyński
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2 Answers2

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Many of the standard examples of non-regular Hausdorff spaces have this property.

For example, there is the space $\mathbb{R}_K$ of Munkres, described here. It is defined to be the real numbers, equipped with the usual topology together with the sets $U\setminus K$, where $U$ is open and $K=\{1,1/2,1/3,\ldots\}$. It is not regular, but every nonempty open set contains an open interval disjoint from $K$, and so we can find a smaller open interval completely inside it.

There is also the philosophically similar slit disc topology on $\mathbb{R}^2$, where we can delete arbitrary subsets of the $y$-axis. It is not regular, but again, every open set contains a disc, so it has the property you want.

Andrew Dudzik
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A very simple example is the one that I described in this answer: it has the desired property, because the isolated points are a dense subset. In fact any non-regular $T_1$ space with a dense set of isolated points has the property.

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Brian M. Scott
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