Continuity of Composite with Inclusion

Theorem

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:

Mapping on Inclusion

If $f$ is $\left({\tau, \tau'}\right)$-continuous, then $f \circ i$ is $\left({\tau_H, \tau'}\right)$-continuous

Inclusion on Mapping

$g$ is $\left({\tau', \tau_H}\right)$-continuous iff $i \circ g$ is $\left({\tau', \tau}\right)$-continuous.

Uniqueness of Induced Topology

The induced topology $\tau_H$ is the only topology on $H$ satisfying Continuity of Composite with Inclusion: Inclusion on Mapping for all possible $g$.