**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 *U*_{i} containing *x* that doesn't intersect it. The intersection of all such *U*_{i} for 1 ≤ *i* ≤ *k* intersected with *V*, is a neighbourhood of *x* that does not intersect the union of this collection of closed sets.

Read more about this topic: Locally Finite Collection

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

**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 ...

... 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 ...

**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 ...

### Famous quotes containing the words sets and/or closed:

“Until, accustomed to disappointments, you can let yourself rule and be ruled by these strings or emanations that connect everything together, you haven’t fully exorcised the demon of doubt that *sets* you in motion like a rocking horse that cannot stop rocking.”

—John Ashbery (b. 1927)

“No other creative field is as *closed* to those who are not white and male as is the visual arts. After I decided to be an artist, the first thing that I had to believe was that I, a black woman, could penetrate the art scene, and that, further, I could do so without sacrificing one iota of my blackness or my femaleness or my humanity.”

—Faith Ringgold (b. 1934)