---
tag: 0PM88Z9W4M
kind: statement
role: proposition
name: Projections are continuous
msc: [54B10]
uses: [0Z57EC6YX3, 0SY9ECQ19Q, 0SQ2PC596Y]
sources:
  - url: https://en.wikipedia.org/wiki/Product_topology
    locator: Properties
---

## Statement

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]] and give $X \times Y$ the [[product topology|0Z57EC6YX3]].
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|0SY9ECQ19Q]].

## 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|0SQ2PC596Y]], 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.