Difference between revisions of "Product topology"
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| − | + | : '''Note: '''{{Note|Very often confused with the [[Box topology]] see [[Product vs box topology]] for details}} | |
| − | + | {{Extra Maths}}__TOC__ | |
| + | ==Definition== | ||
| + | Given an arbitrary collection of [[Indexing set|indexed]] {{M|(X_\alpha,\mathcal{J}_\alpha)_{\alpha\in I} }} [[Topological space|topological spaces]], we define the '''product topology''' as follows: | ||
| + | * Let {{M|1=X:=\prod_{\alpha\in I}X_\alpha}} be a set imbued with the [[Topological space|topology]] generated by the [[Basis (topology)|basis]]: | ||
| + | * {{M|1=\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. | ||
| + | ** {{Yellow Note|For the sake of contrast, the [[Box topology]] has this definition for a basis:<br/><nowiki> | ||
| + | </nowiki>{{M|1=\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 {{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) | ||
| − | <math> | + | ==Characteristic property== |
| − | + | {{Yellow Note|Here {{M|p_i}} denotes the ''[[canonical projection]]'', sometimes {{M|\pi_i}} is used - I avoid using {{M|\pi}} because it is too similar to {{M|\prod}} (at least with my handwriting) - I have seen books using both of these conventions}} | |
| − | + | {{Todo|Finish off}} | |
| − | {{ | + | {| class="wikitable" border="1" |
| + | |- | ||
| + | ! <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 {{M|\forall \alpha\in I}}) | ||
| + | |} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 19:51, 11 October 2015
- Note: Very often confused with the Box topology see Product vs box topology for details
Contents
[hide]Definition
Given an arbitrary collection of indexed (Xα,Jα)α∈I topological spaces, we define the product topology as follows:
- Let X:=∏α∈IXα be a set imbued with the topology generated by the basis:
- B={∏α∈IUα| ∀α∈I[Uα∈Jα]∧∃n∈N[|{Uα|Uα≠Xα}|=n]}
- 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:
Bbox={∏α∈IUα| ∀α∈I[Uα∈Jα]} - the product of any collection of open sets
- Note that in the case of a finite number of spaces, say (Xi,Ji)ni=1 then the topology on ∏ni=1Xi is generated by the basis:
- Bfinite={∏ni=1Ui| ∀i∈{1,2,…,n}[Ui∈Ji]} (that is to say the box/product topologies agree)
Characteristic property
Here pi denotes the canonical projection, sometimes πi is used - I avoid using π because it is too similar to ∏ (at least with my handwriting) - I have seen books using both of these conventions
TODO: Finish off
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| (Commutes ∀α∈I) |
