Closure of Subset of Double Pointed Topological Space

Theorem
Let $\left({S, \tau}\right)$ be a topological space.

Let $D$ be a doubleton endowed with the indiscrete topology.

Let $\left({S \times D, \tau}\right)$ be the double pointed topology on $S$.

Let $X \subseteq S \times D$ be a subset of $S \times D$.

Then the closure of $X$ in $\tau$ is:


 * $\operatorname{cl} \left({X}\right) = \operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right) \times D$

where $\operatorname{pr}_1$ denotes the first projection on $S \times D$.

Proof
By Closed Sets of Double Pointed Topology, $\operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right) \times D$ is closed in $\tau$.

Furthermore, for $\left({s, d}\right) \in X$, one has:


 * $s \in \operatorname{pr}_1 \left({X}\right) \subseteq \operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right)$

by definition of closure.

Since also $d \in D$, we conclude that:


 * $X \subseteq \operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right) \times D$

By $(3)$ of Equivalence of Definitions of Topological Closure, it now suffices to prove that:


 * $\operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right) \times D \subseteq C$

for any closed set $C$ with $X \subseteq C$.

By Closed Sets of Double Pointed Topology, for some $C'$ closed in $\tau$, we have:


 * $C = C' \times D$

so that $C' = \operatorname{pr}_1 \left({C}\right)$.

By Image Preserves Subsets, it follows that:


 * $\operatorname{pr}_1 \left({X}\right) \subseteq C'$

and by $(3)$ of Equivalence of Definitions of Topological Closure, this means:


 * $\operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right) \subseteq C'$

whence by Cartesian Product of Subsets:


 * $\operatorname{cl} \left({\operatorname{pr}_1 \left({X}\right)}\right) \times D \subseteq C' \times D = C$

Hence the result.

Also see

 * Interior in Double Pointed Topology