Open Sets of Double Pointed Topology/Corollary

Theorem

Let $\struct {S, \tau_S}$ 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 $S$.

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 $\map \complement X$ is open.

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

$\map \complement X = U \times D$

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

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$X = \paren {S \times D} \setminus \paren {U \times D} = \paren {S \setminus U} \times D$

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

$\blacksquare$