An Application of Compactness

Section An Application of Compactness

As mentioned at the beginning of this section, compactness is the quality we need to ensure that continuous functions from topological spaces to

R

attain their maximum and minimum values.

In this activity we prove the following theorem.

A continuous function from a compact topological space to the real numbers assumes a maximum and minimum value.

Let

X

be a compact topological space and

f : X \to R

a continuous function. What does the continuity of

f

tell us about

f (X)

R ?

Why can we conclude that the set

f (X)

has a least upper bound

M ?

Why must

M

be an element of

f (X) ?

Complete the proof of Theorem 17.14.

Topology: An Inquiry-Based Approach