Revision as of 19:51, 11 October 2015 (view source) Alec (Talk | contribs) (Not finished) ← Older edit		Revision as of 20:49, 2 May 2016 (view source) Alec (Talk | contribs) m (Last modified in October 2015 for improvement, scrapping all that though. Work started) Newer edit →
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		+	{{Refactor notice|grade=A|As a part of the topology patrol}}
		+	: '''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]].
		+	__TOC__
		+	==Definition==
		+	Given an arbitrary family of [[topological space|topological spaces]], {{M|\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} }} the ''product topology'' is a [[topology]] defined on the set {{M|\prod_{\alpha\in I}X_\alpha}} (where {{M|\prod}} denotes the [[Cartesian product]]) to be the [[topology generated by a basis|topology generated by the basis]]:
		+	* {{MM|1=\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, {{M|1=\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\} }} 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 of the product topology|Characteristic property]]==
		+	{{:Characteristic property of the product topology/Statement}}
		+
		+	=OLD PAGE=
	: '''Note: '''{{Note|Very often confused with the [[Box topology]] see [[Product vs box topology]] for details}}		: '''Note: '''{{Note|Very often confused with the [[Box topology]] see [[Product vs box topology]] for details}}
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	* Note that in the case of a finite number of spaces, say {{M|1=(X_i,\mathcal{J}_i)_{i=1}^n}} then the topology on {{M|1=\prod_{i=1}^nX_i}} is generated by the basis:		* Note that in the case of a finite number of spaces, say {{M|1=(X_i,\mathcal{J}_i)_{i=1}^n}} then the topology on {{M|1=\prod_{i=1}^nX_i}} is generated by the basis:
	** {{M|1=\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\} }} (that is to say the box/product topologies agree)		** {{M|1=\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\} }} (that is to say the box/product topologies agree)
		+
	==Characteristic property==		==Characteristic property==

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Revision as of 20:49, 2 May 2016

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, $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ the product topology is a topology defined on the set $\prod_{\alpha\in I}X_\alpha$ (where $\prod$ 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, $\left\{\pi_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha\text{ given by }\pi_\alpha:(x_\gamma)_{\gamma\in I}\mapsto x_\alpha\ \Big\vert\ \alpha\in I\right\}$ 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

Let $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$ be an arbitrary family of topological spaces and let $(Y,\mathcal{ K })$ be a topological space. Consider $(\prod_{\alpha\in I}X_\alpha,\mathcal{J})$ as a topological space with topology ($\mathcal{J}$) given by the product topology of $\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I}$. Lastly, let $f:Y\rightarrow\prod_{\alpha\in I}X_\alpha$ be a map, and for $\alpha\in I$ define $f_\alpha:Y\rightarrow X_\alpha$ as $f_\alpha=\pi_\alpha\circ f$ (where $\pi_\alpha$ denotes the $\alpha^\text{th}$ canonical projection of the product topology) then:

• $f:Y\rightarrow\prod_{\alpha\in I}X_\alpha$ is continuous

if and only if

• $\forall\beta\in I[f_\beta:Y\rightarrow X_\beta\text{ is continuous}]$ - in words, each component function is continuous

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TODO: Link to diagram

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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 $(X_\alpha,\mathcal{J}_\alpha)_{\alpha\in I}$ topological spaces, we define the product topology as follows:

• Let $X:=\prod_{\alpha\in I}X_\alpha$ be a set imbued with the topology generated by the basis:
• $\mathcal{B}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\wedge\exists n\in\mathbb{N}[\vert\{U_\alpha\vert U_\alpha\ne X_\alpha\}\vert=n]\right\}$
  • 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:
    $\mathcal{B}_\text{box}=\left\{\prod_{\alpha\in I}U_\alpha\Big\vert\ \forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha]\right\}$ - the product of any collection of open sets
• Note that in the case of a finite number of spaces, say $(X_i,\mathcal{J}_i)_{i=1}^n$ then the topology on $\prod_{i=1}^nX_i$ is generated by the basis:
  • $\mathcal{B}_\text{finite}=\left\{\prod^n_{i=1}U_i\Big\vert\ \forall i\in\{1,2,\ldots,n\}[U_i\in\mathcal{J}_i]\right\}$ (that is to say the box/product topologies agree)

Characteristic property

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TODO: Finish off

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