# 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 $g$.