User:Dfeuer/Continuous functions between topological joins

Theorem
Let $X$ and $X'$ be nonempty sets.

Let $\tau_a$ and $\tau'_a$ be topology on $X$ and $X'$ for each $a \in A$.

Let $\tau = \bigvee_{a \in A} \tau_a$.

Let $\tau '= \bigvee_{a \in A} \tau'_a$.

Let $f$ be a mapping from $X$ to $Y$.

Then:


 * If $f$ is $\tau_a$-$\tau'_a$ continuous for each $a$, $f$ is $\tau$-$\tau'$ continuous.

Proof
By some sub-basis criterion for continuity, we need only prove that the inverse image of a member of $\bigcup_{a\in A}\tau_a$ is open.