Category:Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism

This category contains pages concerning Mapping between Subspaces of Homeomorphic Spaces is Homeomorphism:

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: T_1 \to T_2$ be a homeomorphism.

Let $H \subseteq S_1$.

Let $T_H = \struct {H, \tau_H}$ be the topological subspace of $T_1$ under the subspace topology $\tau_H$ induced by $\tau_1$.

Let $K = f \sqbrk H$ be the image of $H$ under $f$.

Let $T_K = \struct {K, \tau_K}$ be the topological subspace of $T_2$ under the subspace topology $\tau_K$ induced by $\tau_2$.

Let $g: H \to K$ be the restriction of $f$ to $H$.

Then $g$ is a homeomorphism.