Continuity_of_Composite_with_Inclusion

Let $T = \left({A, \tau}\right)$ and $T' = \left({A', \tau'}\right)$ be topological spaces. Let $H \subseteq A$. Let $T_H = \left({H, \tau_H}\right)$ be a topological subspace of $T$. Let $i: H \to A$ be the inclusion mapping. Let $f: A \to A'$ and $g: A' \to H$ be mappings. Then the following apply