Compact Sets of Double Pointed Topology/Corollary

Corollary to Compact Sets of Double Pointed Topology

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 $S$.

$\struct {S \times D, \tau}$ is compact if and only if $\struct {S, \tau_S}$ is compact.

Proof

By Projection is Surjection, it follows that:

$\map {\pr_1} {S \times D} = S$

where $\pr_1$ is the first projection on $S \times D$.

The result follows by Compact Sets of Double Pointed Topology.

$\blacksquare$