Closure of Subset of Double Pointed Topological Space

Theorem
Let $\struct {S, \tau}$ be a topological space.

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

Let $\struct {S \times D, \tau}$ 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:


 * $\map \cl X = \map \cl {\map {\pr_1} X} \times D$

where $\pr_1$ denotes the first projection on $S \times D$.

Proof
By Closed Sets of Double Pointed Topology, $\map \cl {\map {\pr_1} X} \times D$ is closed in $\tau$.

Furthermore, for $\tuple {s, d} \in X$, one has:


 * $s \in \map {\pr_1} X \subseteq \map \cl {\map {\pr_1} X}$

by definition of closure.

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


 * $X \subseteq \map \cl {\map {\pr_1} X} \times D$

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


 * $\map \cl {\map {\pr_1} X} \times D \subseteq C$

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 $C' = \map {\pr_1} C$.

By Image Preserves Subsets, it follows that:


 * $\map {\pr_1} X \subseteq C'$

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


 * $\map \cl {\map {\pr_1} X} \subseteq C'$

whence by Cartesian Product of Subsets:


 * $\map \cl {\map {\pr_1} X} \times D \subseteq C' \times D = C$

Hence the result.

Also see

 * Interior in Double Pointed Topology