Continuity of Composite with Inclusion
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Theorem
Let $T = \struct {A, \tau}$ and $T' = \struct {A', \tau'}$ be topological spaces.
Let $H \subseteq A$.
Let $T_H = \struct {H, \tau_H}$ 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 $\tuple {\tau, \tau'}$-continuous, then $f \circ i$ is $\tuple {\tau_H, \tau'}$-continuous
Inclusion on Mapping
$g$ is $\tuple {\tau', \tau_H}$-continuous if and only if $i \circ g$ is $\tuple {\tau', \tau}$-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 $g$.