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 $S \times D$.

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


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

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

Also see

 * Closure in Double Pointed Topology