Open Sets of Double Pointed Topology/Corollary

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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 $S$.


A subset $X \subseteq S \times D$ is closed in $\tau$ if and only if for some closed set $C$ of $\tau$:

$X = C \times D$


Proof

By definition, $X$ is closed if and only if its complement $\complement \left({X}\right)$ is open.

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

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

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 $\tau$, the result follows.

$\blacksquare$