Continuity of Composite with Inclusion/Mapping on Inclusion

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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 $f: A \to A'$ be a mapping.


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


Proof

From Inclusion Mapping is Continuous, $i$ is $\left({\tau_H, \tau}\right)$-continuous.

It follows from Continuity of Composite Mapping that $f \circ i$ is $\left({\tau_H, \tau'}\right)$-continuous.

$\blacksquare$


Sources