Page 133 - DMTH503_TOPOLOGY

Unit 13: Compact Spaces and Compact Subspace of Real Line

It means that {G : 1 i n} is finite subcover of the countable cover Notes
i
{G : n N}
n
Hence Y is countably compact

Self Assessment

1. Prove that a topological space is compact if every basic open cover has a finite sub-cover.
2. Show that every cofinite topological space (X, T) is compact.
3. Show that if (Y, T ) is a compact subspace of a Hausdorff space (X, T), then Y is T-closed.
1
13.2 Compact Subspaces of the Real Line

The theorems of the preceding section enable is to construct new compact spaces from existing
ones, but in order to get very far we have to find some compact spaces to start with. The natural
place to begin is the real line.
Application include the extreme value theorem and the uniform continuity theorem of calculus,
suitably generalised.
Theorem 6: Extreme Value Theorem
Let f: X  Y be continuous, where Y is an ordered set in the order topology. If X is compact, then
there exist points c and d in X such that f(c)  f(x)  f(d) for every x  X.
The extreme value theorem of calculus is the special case of this theorem that occurs when we
take X to be a closed interval in and Y to be .
Proof: Since f is continuous and X is compact, the set A = f(X) is compact. We show that A has a
largest element M and a smallest element m. Then since m and M belong to A, we must have
m = f(c) and M = F(d) for some points c and d of X.
If A has no largest element, then the collection

{(–, a)|a  A}
forms an open covering of A. Since A is compact, some finite subcollection
{(–, a ), ..., (–, a )}
1 n
covers A. If a is the largest of the elements a ,..., a , then a belongs to none of these sets, contrary
i 1 n i
to the fact that they cover A.
A similar argument shows that A has a smallest element.
Definition: Let (X, d) be a metric space; let A be a non-empty subset of X. For each x  X, we
define the distance from x to A by the equation

d(x, A) = inf {d (x, a) | a  A}.
It is easy to show that for fixed A, the function d (x, A) is continuous function of x.

Given x, y X, one has the inequalities
d(x, A) d(x, a) d(x, y) + d(y, a),
for each a  A. It follows that
d(x, A) –d(x, y) inf d (y, a) = d(y, A),

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