Projections are continuous

Statement

Let $X$ and $Y$ be topological spaces and give $X \times Y$ the product topology. Then the projections $\pi_1 \colon X \times Y \to X$, $(x, y) \mapsto x$, and $\pi_2 \colon X \times Y \to Y$, $(x, y) \mapsto y$, are continuous.

Remarks

Proof: for $U$ open in $X$, we have $\pi_1^{-1}(U) = U \times Y$, which is a basic open set. The proof for $\pi_2$ is the same. The projections are also open maps, but in general they are not closed maps. The product topology is the coarsest topology on $X \times Y$ that makes both projections continuous.

Explanations

L1 human

Taking one coordinate of a point is a continuous operation. For example, the map $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is continuous.