Locally Finite Collection - Closed Sets

Closed Sets

It is clear from the definition of a topology that a finite union of closed sets is closed. One can readily give an example of an infinite union of closed sets that is not closed. However, if we consider a locally finite collection of closed sets, the union is closed. To see this we note that if x is a point outside the union of this locally finite collection of closed sets, we merely choose a neighbourhood V of x that intersects this collection at only finitely many of these sets. Define a bijective map from the collection of sets that V intersects to {1, ..., k} thus giving an index to each of these sets. Then for each set, choose an open set Ui containing x that doesn't intersect it. The intersection of all such Ui for 1 ≤ ik intersected with V, is a neighbourhood of x that does not intersect the union of this collection of closed sets.

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Other articles related to "closed sets, set, sets, closed, closed set":

Characterizations Of The Category Of Topological Spaces - Definition Via Closed Sets
... Objects all pairs (X,T) of set X together with a collection T of subsets of X satisfying The empty set and X are in T ... The intersection of any collection of sets in T is also in T ... The union of any pair of sets in T is also in T ...
Borel Set
... In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union ... Borel sets are named after Émile Borel ... For a topological space X, the collection of all Borel sets on X forms a σ-algebra, known as the Borel algebra or Borel σ-algebra ...
More About Closed Sets
... In point set topology, a set A is closed if it contains all its boundary points ... The notion of closed set is defined above in terms of open sets, a concept that makes sense for topological spaces, as well as for other spaces that carry topological structures, such as ... An alternative characterization of closed sets is available via sequences and nets ...

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