Box topology
From Maths
Stub grade: B
This page is a stub
This page is a stub, so it contains little or minimal information and is on a todo list for being expanded.The message provided is:
Not that important because we have the product topology page, however it should be a slightly bigger stub page!
Definition
Let [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath] be an arbitrary family of topological spaces. The box topology is the topology generated by the following basis^{[1]}:
 [ilmath]\mathcal{B}:=\{\prod_{\alpha\in I}U_\alpha\ \vert\ (U_\alpha)_{\alpha\in I}\in\prod_{\alpha\in I}\mathcal{J}_\alpha\}[/ilmath]  the basis consisting of all Cartesian products of all open sets from [ilmath]\big((X_\alpha,\mathcal{J}_\alpha)\big)_{\alpha\in I} [/ilmath]^{[Note 1]}
TODO: Check this expression
Notes
 ↑ Badly phrased, it should mean all tuples of the form [ilmath](U_\alpha)_{\alpha\in I} [/ilmath] where [ilmath]\forall\alpha\in I[U_\alpha\in\mathcal{J}_\alpha][/ilmath]
References
