Nowhere dense subset: Difference between revisions
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A subset of a [[topological space]] is said to be '''nowhere dense''' if the [[interior]] of its [[closure]] is empty. | A subset of a [[topological space]] is said to be '''nowhere dense''' if the [[interior]] of its [[closure]] is empty. | ||
A nowhere dense closed subset is the same as the complement of a dense open subset. In general, | A nowhere dense [[closed subset]] is the same as the complement of a [[dense subset|dense]] [[open subset]]. In general, the complement of a nowhere dense subset is dense, but the complement of a dense subset need not be nowhere dense. | ||
Revision as of 10:08, 18 August 2007
This article defines a property over pairs of a topological space and a subspace, or equivalently, properties over subspace embeddings (viz, subsets) in topological spaces
Definition
Symbol-free definition
A subset of a topological space is said to be nowhere dense if the interior of its closure is empty.
A nowhere dense closed subset is the same as the complement of a dense open subset. In general, the complement of a nowhere dense subset is dense, but the complement of a dense subset need not be nowhere dense.