Finite dimensional continuum with C(X) homeomorphic to Cone(X) must have dimension 1
Theorem
If $Y$ is a finite-dimensional continuum such that $C(X)$ is homeomorphic to $\mathrm{Cone}$$(X)$, then $\mathrm{dim}(X)=1$.
If $Y$ is a finite-dimensional continuum such that $C(X)$ is homeomorphic to $\mathrm{Cone}$$(X)$, then $\mathrm{dim}(X)=1$.