Refinement - Revision history

Vipul: 2 revisions

2008-05-11T19:57:51Z

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Vipul at 06:13, 18 August 2007

2007-08-18T06:13:09Z

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	==Definition==		==Definition==

	Let <math>U_i, i \in I</math> be a cover of a [[topological space]] <math>X</math>, viz a collection of subsets of <math>X</math> whose union is <math>X</math>. A refinement of <math>U_i</math> is another cover <math>V_j, j \in J</math> such that for any <math>j</math>, there exists an <math>i</math> such that <math>V_j \subseteq U_i</math>.		Let <math>U_i, i \in I</math> be a [[cover]] of a [[topological space]] <math>X</math>, viz a collection of subsets of <math>X</math> whose union is <math>X</math>. A refinement of <math>U_i</math> is another cover <math>V_j, j \in J</math> such that for any <math>j</math>, there exists an <math>i</math> such that <math>V_j \subseteq U_i</math>.

	An '''open refinement''' is a refinement where the new cover is an open cover.		An '''open refinement''' is a refinement where the new cover is an open cover.

Vipul at 06:12, 18 August 2007

2007-08-18T06:12:52Z

New page

==Definition==

Let <math>U_i, i \in I</math> be a cover of a [[topological space]] <math>X</math>, viz a collection of subsets of <math>X</math> whose union is <math>X</math>. A refinement of <math>U_i</math> is another cover <math>V_j, j \in J</math> such that for any <math>j</math>, there exists an <math>i</math> such that <math>V_j \subseteq U_i</math>.

An '''open refinement''' is a refinement where the new cover is an open cover.

==Related notions==

* [[Shrinking]]
* [[Expansion]]