Difference between revisions of "Open cover"
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Latest revision as of 11:53, 1 December 2015
Definition
An open cover or open covering is a covering of a set [ilmath]S\subseteq X[/ilmath] consisting of only open sets from the topological space [ilmath](X,\mathcal{J})[/ilmath][1], that is:
- An open covering is a collection [ilmath]\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J} [/ilmath] where [math]S\subseteq\bigcup_{\alpha\in I}U_\alpha[/math]
Note on definition
This is exactly what "an open covering of a set" means. However the definition is usually involved in defining compactness so one may see the definition:
- [ilmath]X=\bigcup_{\alpha\in I}U_\alpha[/ilmath] instead (notice the strict equality) and it not be defined for subsets.
As the compactness page shows, it actually doesn't matter if you talk about compactness of subsets or compactness of subspaces. As such the intuitive definition of a covering is fine, something that "covers" something else, it need not be equal to it.
- TL;DR - an open covering is a covering by open sets. Nothing more
