Interior of Subset of Double Pointed Topological Space

Theorem
Let $\left({S, \vartheta}\right)$ 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 $S \times D$.

Define $A \subseteq S$ by:


 * $A := \left\{{s \in S }\,\middle\vert\,{ \forall d \in D: \left({s, d}\right) \in X}\right\}$

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


 * $X^\circ = A^\circ \times D$

Also see

 * Closure in Double Pointed Topology