Locally closed subset (original) (raw)
In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if the following equivalent conditions are met: * E is an intersection of an open subset and a closed subset of X. * For each point x in E, there is a neighborhood U of x in X such that is closed in U. * E is an open subset of the closure . A topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion.
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| dbo:abstract | In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if the following equivalent conditions are met: * E is an intersection of an open subset and a closed subset of X. * For each point x in E, there is a neighborhood U of x in X such that is closed in U. * E is an open subset of the closure . The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed. To see the second condition implies the third, use the facts that for subsets , A is closed in B if and only if and that for a subset E and an open subset U, . Finite intersections and the pre-image under a continuous map of locally closed sets are locally closed. On the other hand, a union and a complement of locally closed subsets need not be locally closed. (This motivates the notion of a constructible set.) For example, is a locally closed subset of . For another example, consider the relative interior D of a closed disk in . It is locally closed since it is an intersection of the closed disk and an open ball. Recall that, by definition, a submanifold E of an n-manifold M is a subset such that for each point x in E, there is a chart around it such that . Hence, a submanifold is locally closed. For a locally closed subset E, the complement is called the boundary of E (not to be confused with topological boundary). If E is a closed submanifold-with-boundary of a manifold M, then the relative interior (i.e., interior as a manifold) of E is locally closed in M and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset. A topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion. (en) |
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| rdfs:comment | In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if the following equivalent conditions are met: * E is an intersection of an open subset and a closed subset of X. * For each point x in E, there is a neighborhood U of x in X such that is closed in U. * E is an open subset of the closure . A topological space is said to be submaximal if every subset is locally closed. See Glossary of topology#S for more of this notion. (en) |
| rdfs:label | Locally closed subset (en) |
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