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