Refinement: Difference between revisions
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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. | ||
Revision as of 06:13, 18 August 2007
Definition
Let be a cover of a topological space , viz a collection of subsets of whose union is . A refinement of is another cover such that for any , there exists an such that .
An open refinement is a refinement where the new cover is an open cover.