# 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.

### 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 ...