General Topology - Finite Product Topology

I hate topology. --- By Woziji Shuode

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Please read to "General Topology - Topology and Basis" first.

The Product Topology on $X \times Y$

Definition 1. [Product Topology]

Let $X$ and $Y$ be topological spaces, then $\mathfrak{B} = \{U \times V\}$ is a basis of $X \times Y, while $U$ is open in $X$ and $V$ is open in $Y$.

Definition 2. [Projection]

Theorem 3.

If $\mathfrak{B}$ is a basis of $X$ and $\mathfrak{C}$ is a basis of $Y$, then $\mathcal{D} = \{B \times C: B \in \mathfrak{B},\;C \in \mathfrak{C}\}$ is a basis for the topology of $X \times Y$.

Theorem 4.

$\mathcal{S} = \{\pi_1^{-1}(U)\;|\;U\;open\;in\;X\}\cup\{\pi_2^{-1}(V)\;|\;V\;open\;in\;Y\}$ is a subbasis for the product topology on $X \times Y$.