Sierpinski space and the particular-point topology, with their closures and their continuous maps

Let $X$ be a set, let $p \in X$, and give $X$ the particular-point topology $\mathcal{T}_p = \{\varnothing\} \cup \{\, U \subseteq X : p \in U \,\}$ (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies). Then, for $A \subseteq X$:

1. Closed sets. The closed subsets of $(X, \mathcal{T}_p)$ are $X$ together with the sets not containing $p$.
2. Interior and closure (Interior, closure, boundary, exterior, derived set and isolated point in a topological space): $\operatorname{int}(A) = \begin{cases} A & p \in A \\ \varnothing & p \notin A, \end{cases} \qquad \overline{A} = \begin{cases} X & p \in A \\ A & p \notin A. \end{cases}$ In particular $\{p\}$ is dense (Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets) and is the smallest dense subset, while $X \setminus \{p\}$ has empty interior.
3. No two nonempty open sets are disjoint, since every nonempty open set contains $p$.
4. Sierpinski space is the case of a two-point set. With $S = \{a,b\}$, $a \ne b$, and particular point $b$, the topology is $\mathcal{T}_{\mathrm{Sier}} = \{\varnothing, \{b\}, S\}$; here $\overline{\{b\}} = S$ and $\overline{\{a\}} = \{a\}$, so $b$ is the open point and $a$ the closed point.
5. Continuous maps into Sierpinski space are exactly the open subsets of the source. For a topological space $Y$, the assignment $f \mapsto f^{-1}[\{b\}]$ is a bijection from the set of continuous maps $Y \to (S, \mathcal{T}_{\mathrm{Sier}})$ onto the topology of $Y$ (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f(\overline{A}) \subseteq \overline{f(A)}$).

• Sierpinski space is the smallest space that is not indiscrete and not discrete, and claim 5 is why it matters: it represents the notion "open set" as a mapping problem, in the same way that a two-element set represents "subset". Every topology on $Y$ is recovered as the set of continuous maps $Y \to S$.

• When $X$ has at least two points, the particular-point topology separates distinct points only in the weakest sense. Any two distinct points are distinguished by an open set: if one of them is $p$, then $\{p\}$ is open and contains $p$ but not the other; and if neither is $p$, then $\{x,p\}$ is open and contains $x$ but not $y$. It is not Hausdorff: every two nonempty open sets meet at $p$, and $\{p\}$ is dense by claim 2. When $X=\{p\}$, by contrast, the topology is discrete and the separation axioms hold vacuously. In every case the space is first countable, since $\{\{x,p\}\}$ is a one-element neighbourhood base at $x$: any neighbourhood of $x$ contains an open set containing $x$, which contains $p$ as well.

• A comparison with the cofinite topology. When $X$ is infinite, both have the property that any two nonempty open sets meet (On an infinite set the cofinite topology has every infinite subset dense and no two nonempty open sets disjoint), but the particular-point topology achieves it with a single point doing all the work. When $X$ has at least two points, its particular point $p$ is not closed, whereas every point is closed in the cofinite topology.