Projection from Product Topology is Continuous

Theorem
Let $$T_1 = \left({A_1, \vartheta_1}\right)$$ and $$T_2 = \left({A_2, \vartheta_2}\right)$$ be topological spaces.

Let $$T = T_1 \times T_2$$ be the topological product of $$T_1$$ and $$T_2$$.

Let $$\operatorname{pr}_1: T \to T_1$$ and $$\operatorname{pr}_2: T \to T_2$$ be the first and second projections from $$T$$ onto its factors.

Then both $$\operatorname{pr}_1$$ and $$\operatorname{pr}_2$$ are continuous.

Proof
If $$U$$ is open in $$T_1$$ then $$\operatorname{pr}_1^{-1} \left({U}\right) = U \times T_2$$ is one of the open sets in the basis in the definition of Product Topology.

Thus $$\operatorname{pr}_1$$ is continuous.

The same argument applies to $$\operatorname{pr}_2$$.