Compact Sets of Double Pointed Topology/Corollary

Corollary to Compact Sets of Double Pointed Topology
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 $S$.

Then $\left({S \times D, \tau}\right)$ is compact iff $\left({S, \vartheta}\right)$ is compact.

Proof
By Projections are Surjections, it follows that:


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

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

The result follows by Compact Sets of Double Pointed Topology.