Cone on Compact Space is Compact

Theorem

Let $A$ be a compact topological space.

Let $C A$ denote the cone on $A$.

Then, $C A$ is compact.

Proof

By definition of cone:

$C A = T \ast A$

where:

$T$ denotes the trivial topological space

$T \ast A$ denotes the join of $T$ and $A$

By Finite Topological Space is Compact, $T$ is compact.

Therefore, by Join of Compact Spaces is Compact:

$C A$ is compact.

$\blacksquare$