Open Sets of Double Pointed Topology/Corollary

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 $S$.

A subset $X \subseteq S \times D$ is closed in $\tau$ iff for some closed set $C$ of $\vartheta$:


 * $X = C \times D$

Proof
By definition, $X$ is closed iff its complement $\complement \left({X}\right)$ is open.

By Open Sets of Double Pointed Topology, it follows by that for some $U \in \vartheta$:


 * $\complement \left({X}\right) = U \times D$

Then by Cartesian Product Distributes over Set Difference and Complement of Complement, we have that:


 * $X = \left({S \times D}\right) \setminus \left({U \times D}\right) = \left({S \setminus U}\right) \times D$

Since $S \setminus U$ is a closed set of $\vartheta$, the result follows.