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
^ abBourbaki 2007, Ch. 1, § 3, no. 3.
^Pflaum 2001, Explanation 1.1.2.
^Ganster, M.; Reilly, I. L. (1989). "Locally closed sets and LC -continuous functions". International Journal of Mathematics and Mathematical Sciences. 12 (3): 417–424. doi:10.1155/S0161171289000505. ISSN 0161-1712.
^Engelking 1989, Exercise 2.7.1.
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