7 Compactness

Question 1

A compact subset, compactness

Answer 1

Let A be a subset of a topological space (X, T). Then A is said to be compact if for every set I and every family of open sets, O_i, i e I, such that A (_ U_{i e I} O_i there exists a finite subfamily O_i1, O_i2,..,O_in such that A (_ O_i1 U O_i2 U … U O_in.

Question 2

Covering (cover), Open covering, finite subcovering

Answer 2

Let I be a set and O_i, i e I, a family of subsets of X. Let A be a subset of X. Then O_i, i e I, is said to be a covering of X if A (_ U_ieI Oi. If each O_i, i e I, is an open set in (X, T), then O_i, i e I is said to be an open covering of A if A (_ U_ieI O_i. A finite subfamily, O_i1, O_i2,…, O_in, of O_i, i e I, is called a finite subcovering (of A) if A (_ O_i1 U O_i2 U…

Question 3

Compactness, definition using coverings

Answer 3

A subset A of a topological space (X, T) is said to be compact if every open covering of A has a finite subcovering. If the compact subset A equals X, then (X, T) is said to be a compact space.

Question 4

Bounded subset of a metric space

Answer 4

A subset A of a metric space (X, d) is said to be bounded if there exists a real number r such that d(a1, a2) <= r , for all a1 and a2 in A.