Compact Convex Sets with Nonempty Interior are Homeomorphic
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Theorem
Let $n \in \N_{> 0}$.
Let $T, T' \subseteq \R^n$ be compact convex subsets of real Euclidean $n$-space.
Then, $T$ is homeomorphic to $T'$.
Proof
By Boundary of Compact Convex Set with Nonempty Interior is Homeomorphic to Sphere:
- $\partial T \sim \Bbb S^{n - 1}$
- $\partial T' \sim \Bbb S^{n - 1}$
Thus, by Homeomorphism Relation is Equivalence:
- $\partial T \sim \partial T'$
Hence, by Cones on Homeomorphic Spaces are Homeomorphic:
- $C \partial T \sim C \partial T'$
But, by Compact Convex Set with Nonempty Interior is Homeomorphic to Cone on Boundary:
- $T \sim C \partial T$
- $T' \sim C \partial T'$
Therefore, by Homeomorphism Relation is Equivalence:
- $T \sim T'$
$\blacksquare$