Difference between revisions of "Covering"

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(Created page with " ==Definition== A covering of a set {{m|A}} is a collection {{m|\mathcal{A} }} where <math>A\subset\cup_{S\in\mathcal{A}}S</math>, that is as you'd expect, a collection of se...")
 
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==Definition==
 
==Definition==
 
A covering of a set {{m|A}} is a collection {{m|\mathcal{A} }} where <math>A\subset\cup_{S\in\mathcal{A}}S</math>, that is as you'd expect, a collection of sets which contain {{m|A}} in their union.
 
A covering of a set {{m|A}} is a collection {{m|\mathcal{A} }} where <math>A\subset\cup_{S\in\mathcal{A}}S</math>, that is as you'd expect, a collection of sets which contain {{m|A}} in their union.

Revision as of 03:06, 7 June 2015

Definition

A covering of a set A is a collection A where ASAS, that is as you'd expect, a collection of sets which contain A 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 X 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.