Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous

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Theorem

Let $\struct{X, \tau_1}$ and $\struct{Y, \tau_2}$ be topological spaces.

Let $f : \struct{X, \tau_1} \to \struct{Y, \tau_2}$ be a continuous mapping.


Let $\tau'_1$ be a finer topology on $X$ than $\tau_1$, that is, $\tau_1 \subseteq \tau'_1$.

Let $\tau'_2$ be a coarser topology on $Y$ than $\tau_2$, that is, $\tau'_2 \subseteq \tau_2$.


Then:

$f : \struct{X, \tau'_1} \to \struct{Y, \tau'_2}$ is a continuous mapping.

Proof

Let $U \in \tau'_2$.


Since $\tau'_2$ is a coarser topology than $\tau_2$:

$U \in \tau_2$.

By definition of continuity:

$\map {f^{-1}} U \in \tau_1$.

Since $\tau'_1$ is a finer topology than $\tau_1$:

$\map {f^{-1}} U \in \tau'_1$.


Since $U$ was arbitrary, by definition of continuity:

$f : \struct{X, \tau'_1} \to \struct{Y, \tau'_2}$ is a continuous mapping

$\blacksquare$