An Application of Compactness

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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.

Activity 17.7.

In this activity we prove the following theorem.

Theorem 17.14.

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

(a)

Let $X$ be a compact topological space and $f: X \to \R$ a continuous function. What does the continuity of $f$ tell us about $f(X)$ in $\R\text{?}$

(b)

Why can we conclude that the set $f(X)$ has a least upper bound $M\text{?}$ Why must $M$ be an element of $f(X)\text{?}$

(c)

Complete the proof of Theorem 17.14.

Topology: An Inquiry-Based Approach

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Section An Application of Compactness

Activity 17.7.

Theorem 17.14.

(a)

(b)

(c)