Continuity Test using Basis

Theorem
Let $T_1, T_2$ be topological spaces.

Let $f: T_1 \to T_2$ be a mapping.

Let $\mathcal B$ be a basis for $T_2$.

In order to determine whether $f: T_1 \to T_2$ is continuous, it is sufficient to prove that $\forall B \in \mathcal B: f^{-1} \left({B}\right)$ is open in $T_1$.

Proof
Suppose it has been proved that for all $\forall B \in \mathcal B$, $f^{-1} \left({B}\right)$ is open in $T_1$.

Let $U$ be open in $T_2$.

From the definition of basis, it follows that $U$ is of the form $\displaystyle \bigcup_I B_i$, where $\forall i \in I: B_i \in \mathcal B$.

Hence:

As $\displaystyle \bigcup_I \left({f^{-1} \left({B_i}\right)}\right)$ is the union of sets which by hypothesis are open in $T_1$, it follows that $f^{-1} \left({U}\right)$ is open in $T_1$.

The result follows from the definition of continuity.