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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#REDIRECT [[Cover]]
 
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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.
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Now {{C|<nowiki>[[covering]]</nowiki>}} can be used, rather than the longer {{C|<nowiki>[[cover|covering]]</nowiki>}}
 
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==Alternative statement==
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Munkres seems to go a different route and only lets one cover entire [[Topological space|spaces]], not sets within it. However he shows that considering any set as a [[Subspace topology|subspace]] of {{m|X}} we can then cover it using (open, as it is brought up studying compactness) sets in the ambient space.
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This is mentioned, discussed and proven on the [[Compactness|compactness]] page.
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{{Definition|Topology}}
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{{Definition|Topology|Metric Space}}

Latest revision as of 15:15, 2 December 2015

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Definition

Now [[covering]] can be used, rather than the longer [[cover|covering]]