Box topology

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Not that important because we have the product topology page, however it should be a slightly bigger stub page!


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


  1. 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]


  1. Introduction to Topological Manifolds - John M. Lee