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 $$\bigcup_I B_i$$, where $$\forall i \in I: B_i \in \mathcal{B}$$.

Hence:

$$ $$

As $$\bigcup_I \left({f^{-1} \left({B_i}\right)}\right)$$ 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.