Continuity of Composite with Inclusion/Mapping on Inclusion

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

If $f$ is $\tuple {\tau, \tau'}$-continuous, then $f \circ i$ is $\tuple {\tau_H, \tau'}$-continuous

Proof

From Inclusion Mapping is Continuous, $i$ is $\tuple {\tau_H, \tau}$-continuous.

It follows from Composite of Continuous Mappings is Continuous that $f \circ i$ is $\tuple {\tau_H, \tau'}$-continuous.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.4$: Subspaces: Proposition $3.4.2 \ \text{(a)}$