Cones on Homeomorphic Spaces are Homeomorphic

Theorem

Let $X, Y$ be topological spaces.

Suppose:

$X \sim Y$

denoting that $X$ is homeomorphic to $Y$.

Then:

$C X \sim C Y$

where $C X$ denotes the cone on $X$.

Proof

Let $T$ be the trivial topological space used in the definition of cone.

We have:

$T \sim T$ by Homeomorphism Relation is Equivalence

$X \sim Y$ by hypothesis

Then, by Joins of Homeomorphic Spaces are Homeomorphic:

$C X \sim C Y$

$\blacksquare$