Proposition

If $X_1, \cdots, X_n$ are topological spaces, each canonical projection $\pi_i: X_1 \times \cdots \times X_n \rightarrow X_i$ is continuous.

Solution

We will use the Characteristic Property of the Product Topology (Theorem 3.27, Introduction to Topological Manifolds). The identity function $f: X_1 \times \cdots \times X_n \rightarrow X_1 \times \cdots \times X_n$ is continuous because any identity function is continuous.

By the Characteristic Property of the Property Topology, $f$’s continuity implies the continuity of $\pi_i \circ f$ for each $i$. Since $f$ is the identity map, $\pi_i \circ f = \pi_i$ for each $i$. Therefore, each $\pi_i$ is continuous.