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Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator C such that () is the intersection of the closed sets containing X. This equivalence remains true for partially ordered sets with the greatest-lower-bound property , if one replace "closed sets" by "closed elements" and ...
These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following. In any discrete space, since every set is closed (and also open), every set is equal to its closure.
The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense. Singleton points (and thus finite sets) are closed in T 1 spaces and Hausdorff spaces. The set of integers is an infinite and unbounded closed set in the real numbers.
Closure operators are determined by their closed sets, i.e., by the sets of the form cl(X), since the closure cl(X) of a set X is the smallest closed set containing X. Such families of "closed sets" are sometimes called closure systems or "Moore families". [1] A set together with a closure operator on it is sometimes called a closure space.
For another example, consider the relative interior of a closed disk in . It is locally closed since it is an intersection of the closed disk and an open ball. On the other hand, { ( x , y ) ∈ R 2 ∣ x ≠ 0 } ∪ { ( 0 , 0 ) } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\neq 0\}\cup \{(0,0)\}} is not a locally closed subset of R 2 ...
As discussed above, given a topological space we may define the closure of any subset to be the set () = {|}, i.e. the intersection of all closed sets of which contain . The set c ( A ) {\displaystyle \mathbf {c} (A)} is the smallest closed set of X {\displaystyle X} containing A {\displaystyle A} , and the operator c : ℘ ( X ) → ℘ ( X ...
Cut locus C(P) of a point P on the surface of a cylinder. A point Q in the cut locus is shown with two distinct shortest paths , connecting it to P.. In the Euclidean plane, a point p has an empty cut locus, because every other point is connected to p by a unique geodesic (the line segment between the points).
The derived set of a subset of a space need not be closed in general. For example, if = {,} with the trivial topology, the set = {} has derived set ′ = {}, which is not closed in . But the derived set of a closed set is always closed.