Difference between revisions of "Covering"
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==Definition== | ==Definition== | ||
| − | A covering of a set {{m|A}} is a collection {{m|\mathcal{A} }} where < | + | A covering of a set {{m|A}} is a collection {{m|\mathcal{A} }} where <mm>A\subseteq\bigcup_{S\in\mathcal{A}}S</mm>, that is as you'd expect, a collection of sets which contain {{m|A}} in their union. |
==Alternative statement== | ==Alternative statement== | ||
Revision as of 03:35, 22 June 2015
Definition
A covering of a set [ilmath]A[/ilmath] is a collection [ilmath]\mathcal{A} [/ilmath] where [math]A\subseteq\bigcup_{S\in\mathcal{A}}S[/math], that is as you'd expect, a collection of sets which contain [ilmath]A[/ilmath] in their union.
Alternative statement
Munkres seems to go a different route and only lets one cover entire spaces, not sets within it. However he shows that considering any set as a subspace of [ilmath]X[/ilmath] we can then cover it using (open, as it is brought up studying compactness) sets in the ambient space.
This is mentioned, discussed and proven on the compactness page.
