Product topology
From Maths
- Note: Very often confused with the Box topology see Product vs box topology for details
Contents
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
| [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]) |
