In
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of mathematics, a subset
of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is said to be locally closed if any of the following equivalent conditions are satisfied:
*
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 an open subset of 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.
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
Examples
The interval