# 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

# 2nd generation page

Note: for finite collections of topological spaces the product and box topology agree. In general however the box topology does not satisfy the characteristic property of the product topology.

## Definition

Given an arbitrary family of topological spaces, [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] the product topology is a topology defined on the set [ilmath]\prod_{\alpha\in I}X_\alpha[/ilmath] (where [ilmath]\prod[/ilmath] denotes the Cartesian product) to be the topology generated by the basis:

• $\mathcal{B}:=\left\{\left.\prod_{\alpha\in I}U_\alpha\right\vert\ (U_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}\mathcal{J}_\alpha\ \wedge\ \Big\vert\{U_\alpha\vert\ U_\alpha\ne X_\alpha\}\Big\vert\in\mathbb{N}\right\}$

The family of functions, [ilmath]\left\{\pi_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha\text{ given by }\pi_\alpha:(x_\gamma)_{\gamma\in I}\mapsto x_\alpha\ \Big\vert\ \alpha\in I\right\}[/ilmath] are called the canonical projections for the product.

Claim 1: this is a basis for a topology,
Claim 2: the canonical projections are continuous

## 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

# OLD PAGE

Note: Very often confused with the Box topology see Product vs box topology for details

$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$

## Definition

Given an arbitrary collection of indexed [ilmath](X_\alpha,\mathcal{J}_\alpha)_{\alpha\in I} [/ilmath] topological spaces, we define the product topology as follows:

• Let [ilmath]X:=\prod_{\alpha\in I}X_\alpha[/ilmath] be a set imbued with the topology generated by the basis:
• [ilmath]\mathcal{B}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\wedge\exists n\in\mathbb{N}[\vert\{U_\alpha\vert U_\alpha\ne X_\alpha\}\vert=n]\right\}[/ilmath]
• That is to say the basis set contains all the products of open sets where the product has a finite number of elements that are not the entirety of their space.
• For the sake of contrast, the Box topology has this definition for a basis:
[ilmath]\mathcal{B}_\text{box}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\right\}[/ilmath] - the product of any collection of open sets
• Note that in the case of a finite number of spaces, say [ilmath](X_i,\mathcal{J}_i)_{i=1}^n[/ilmath] then the topology on [ilmath]\prod_{i=1}^nX_i[/ilmath] is generated by the basis:
• [ilmath]\mathcal{B}_\text{finite}=\left\{\prod^n_{i=1}U_i\Big\vert\ \forall i\in\{1,2,\ldots,n\}[U_i\in\mathcal{J}_i]\right\}[/ilmath] (that is to say the box/product topologies agree)

## Characteristic property

Here [ilmath]p_i[/ilmath] denotes the canonical projection, sometimes [ilmath]\pi_i[/ilmath] is used - I avoid using [ilmath]\pi[/ilmath] because it is too similar to [ilmath]\prod[/ilmath] (at least with my handwriting) - I have seen books using both of these conventions

TODO: Finish off

$\begin{xy} \xymatrix{ & \prod_{\beta\in I}X_\beta \ar[d]^{p_i} \\ Y \ar[ur]^f \ar[r]_{f_i} & X_i }\end{xy}$
(Commutes [ilmath]\forall \alpha\in I[/ilmath])