Join of Compact Spaces is Compact

Theorem

Let $A, B$ be compact topological spaces.

Then:

$A \ast B$ is compact

where $A \ast B$ denotes the join of $A$ and $B$.

Proof

By Closed Real Interval is Compact:

$\closedint 0 1$

is compact.

Thus, by Topological Product of Compact Spaces:

$A \times B \times \closedint 0 1$

is compact.

Therefore, by Quotient Space of Compact Space is Compact:

$A \ast B = \paren {A \times B \times \closedint 0 1} / \RR$

is compact.

$\blacksquare$