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 $$pr_1: T \to T_1$$ and $$pr_2: T \to T_2$$ be the first and second projections from $$T$$ onto its factors.

Then both $$pr_1$$ and $$pr_2$$ are continuous.

Proof
If $$U$$ is open in $$T_1$$ then $$pr_1^{-1} \left({U}\right) = U \times T_2$$ is one of the open sets in the basis in the definition of Product Topology.

Thus $$pr_1$$ is continuous.

The same argument applies to $$pr_2$$.