If $K : D(K)\to Y $ is a continuous one-to-one operator and C is a subspace, $D(K)$ is a compact set, then the inverse operator is continuous.

Question

If $K : D(K)\to Y $
is a continuous one-to-one operator and $C$ is a subspace and $D(K)$ is a compact set, then the inverse operator $(K\mid C)^{-1}$ is continuous? Here, $K\mid C$ denotes the restriction of $K$ to $C$ and $D(K)$ is the domain of $K$.

Answer 1

Let $f:C→R(f)$ be a continuous bijection from a compact $C$ to a Hausdorff space, then it is a homeomorphism.

https://proofwiki.org/wiki/Continuous_Bijection_from_Compact_to_Hausdorff_is_Homeomorphism