Product Topology

Definition

A product topology is the Cartesian product of a family of topological spaces

Let $i\in I$ be an index set, and $X_i$ be a Topological Space. The product topology is defined as $X:={\prod }_{i\in I}{X}_i=\left\{x:I\to {\bigcup }_{i\in I}{X}_i|\forall i\in I,x_i\in X_i\right\}$ where $x_i$ is the $i$-th coordinate (component) of the function $x$.

Facts

Consider bases ${\mathcal{B}}_X,{\mathcal{B}}_Y$ for the topologies on $X,Y$. Then, ${\mathcal{B}}_{X\times Y}=\{B_X\times B_Y|B_X\in {\mathcal{B}}_Y,B_Y\in {\mathcal{B}}_Y\}$ is a basis for the topology on $X\times Y$.

The countably infinite product space ${\mathbb{R}}^{\omega }=\prod _{n\in \mathbb{Z}}\mathbb{R}$ of real numbers with the Product Topology is metrizable.

Consider a function $f:Y\to \prod _{i\in I}{X}_i$ between a Topological Space and a product space. Then, the function $f$ is continuous if and only if each Composite Function ${\pi }_i\circ f$ of the function $f$ and the Projection Map ${\pi }_i$ is continuous. $f\in C^0⟺\forall i\in I,{\pi }_i\circ f\in C^0$

The Product Space of any collection of Hausdorff spaces is a Hausdorff space.

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The Product Space of any collection of connected spaces is a connected space.

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The Product Space of two separable spaces is a separable space.

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The Product Space of two first-countable spaces is a first-countable space.

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The Product Space of two second-countable spaces is a second-countable space.

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Tychonoff's Theorem

Definition

The Product Topology of any collection of compact topological spaces is compact.

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