Urysohn's lemma and arcwise connected completely distributive lattices (Q2725211)
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scientific article; zbMATH DE number 1618989
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Urysohn's lemma and arcwise connected completely distributive lattices |
scientific article; zbMATH DE number 1618989 |
Statements
28 August 2002
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completely distributive lattice
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interval topology
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normal space
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Uryson's lemma
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0.9174875
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0.8661168
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0.86438954
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Urysohn's lemma and arcwise connected completely distributive lattices (English)
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Suppose \(L\) is a completely distributive lattice equipped with the interval topology. The authors prove that the following conditions are equivalent: NEWLINENEWLINENEWLINE(1) \(L\) satisfies Uryson condition, that is to say, for any two disjoint closed sets \(A\) and \(B\) in a normal space \(X\), there is a continuous function \(f: X\rightarrow L\) such that \(f(A) = \{0\}\) and \(f(B) = \{1\}\). (2) \(L\) is arcwise connected. (3) \(L\) contains a maximal chain which is isomorphic to the unit interval [0, 1].
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