Product topology
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Contents
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.
 [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
We claim:
 [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
 this topology is the unique topology on [ilmath]X[/ilmath] for which the characteristic property (see below) holds
Characteristic property
 [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
Notes
References

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:
 [math]\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\}[/math]
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
 [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
OLD PAGE
 Note: Very often confused with the Box topology see Product vs box topology for details
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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
TODO: Finish off
[math]\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} [/math] 

(Commutes [ilmath]\forall \alpha\in I[/ilmath]) 