Locally closed subset
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In topology, a branch of mathematics, a subset of a topological space is said to be locally closed if any of the following equivalent conditions are satisfied:[1][2][3][4]
- is the intersection of an open set and a closed set in
- For each point there is a neighborhood of such that is closed in
- is open in its closure
- The set is closed in
- is the difference of two closed sets in
- is the difference of two open sets in
The second condition justifies the terminology locally closed and is Bourbaki's definition of locally closed.[1] To see the second condition implies the third, use the facts that for subsets is closed in if and only if and that for a subset and an open subset