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Product space

Definition

Given two topological spaces $(X,\mathcal{T}_X), (Y,\mathcal{T}_Y)$, their topological product is the Cartesian product $X \times Y$ with the topology with basis $\{U \times V: U \in \mathcal{T}_X, V \in \mathcal{T}_Y\}$.

Properties

The canonical projection maps $p_X: X \times Y \to X, (x,y) \to x$ and $p_Y: X \times Y \to Y, (x,y) \to y$ are continuous maps.

The product topology is the coarsest topology such that the projection maps are continuous.

$X \times Y$ is connected if and only if $X$ and $Y$ are connected.

$X \times Y$ is compact if and only if $X$ and $Y$ are compact.

all these pages adapted with probably insufficient credit from my university's lecture notes