# Product topology

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As a part of the topology patrol
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Check Munkres and Topological Manifolds

## Definition

Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}[/ilmath] be an arbitrary family of topological spaces. The product topology is a new topological space defined on the set [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] (herein we define [ilmath]X:=\prod_{\alpha\in I}X_\alpha[/ilmath] for notational convenience, where [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] denotes the Cartesian product of the family [ilmath](X_\alpha)_{\alpha\in I}[/ilmath]) with topology, [ilmath]\mathcal{J} [/ilmath] defined as:

• the topology generated by the basis [ilmath]\mathcal{B} [/ilmath], where [ilmath]\mathcal{B} [/ilmath] is defined as follows:
• [ilmath]\mathcal{B}:=\left.\left\{\prod_{\alpha\in I}U_\alpha\ \right\vert\ (\forall\beta\in I[U_\beta\in\mathcal{J}_\beta])\wedge\vert\{U_\alpha\ \vert\ \alpha\in I\wedge U_\alpha\neq X_\alpha\}\vert\in\mathbb{N}\right\}[/ilmath] Caution:I need to check this expression
• In words, [ilmath]\mathcal{B} [/ilmath] is the set that contains all Cartesian products of open sets, [ilmath]U_\alpha\in\mathcal{J}_\alpha[/ilmath] given only finitely many of those open sets are not equal to [ilmath]X_\alpha[/ilmath] itself.

We claim:

1. [ilmath]\mathcal{B} [/ilmath] satisfies the conditions for a topology to be generated by a basis, thus yielding a topology on [ilmath]X[/ilmath], and
2. this topology is the unique topology on [ilmath]X[/ilmath] for which the characteristic property (see below) holds

## Characteristic property

 TODO: Caption [ilmath]\begin{xy} \xymatrix{ & & \prod_{\alpha\in I}X_\alpha \ar[dd] \\ & & \\ Y \ar[uurr]^f \ar[rr]+<-0.9ex,0.15ex>|(.875){\hole} & & X_b \save (15,13)+"3,3"*+{\ldots}="udots"; (8.125,6.5)+"3,3"*+{X_a}="x1"; (-8.125,-6.5)+"3,3"*+{X_c}="x3"; (-15,-13)+"3,3"*+{\ldots}="ldots"; \ar@{->} "x1"; "1,3"; \ar@{->}_(0.65){\pi_c,\ \pi_b,\ \pi_a} "x3"; "1,3"; \ar@{->}|(.873){\hole} "x1"+<-0.9ex,0.15ex>; "3,1"; \ar@{->}_{f_c,\ f_b,\ f_a} "x3"+<-0.9ex,0.3ex>; "3,1"; \restore } \end{xy}[/ilmath]
Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces and let [ilmath](Y,\mathcal{ K })[/ilmath] be a topological space. Consider [ilmath](\prod_{\alpha\in I}X_\alpha,\mathcal{J})[/ilmath] as a topological space with topology ([ilmath]\mathcal{J} [/ilmath]) given by the product topology of [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath]. Lastly, let [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] be a map, and for [ilmath]\alpha\in I[/ilmath] define [ilmath]f_\alpha:Y\rightarrow X_\alpha[/ilmath] as [ilmath]f_\alpha=\pi_\alpha\circ f[/ilmath] (where [ilmath]\pi_\alpha[/ilmath] denotes the [ilmath]\alpha^\text{th} [/ilmath] canonical projection of the product topology) then:
• [ilmath]f:Y\rightarrow\prod_{\alpha\in I}X_\alpha[/ilmath] is continuous
• [ilmath]\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}][/ilmath] - in words, each component function is continuous

TODO: Link to diagram