Compactness and Continuity

Section Compactness and Continuity

In our preview activity we learned about compactness — the analog of closed intervals from

R

in topological spaces. Recall that a subset

A

of a topological space

X

is compact if every open cover of

A

has a finite sub-cover. As we will see, the definition of compactness is exactly what we need to ensure results of the type that continuous real-valued functions with domains in topological spaces attain maximum and minimum values on compact sets.

🔗

We might expect that compact sets have certain properties, but we must be careful which ones we assume.

🔗

Activity 17.2.

🔗

Let

X = {a, b, c, d}

and give

X

the topology

τ = {\emptyset, {a}, {b, c}, {a, b, c}, X} .

🔗

(a)

🔗

Explain why every finite subset of a topological space must be compact.

🔗

(b)

🔗

Find, if possible, a subset of

X

that is compact and open. If no such subset exists, explain why.

🔗

(c)

🔗

A

is a compact subset of

X,

must

A

be open? Explain.

🔗

(d)

🔗

Find, if possible, a subset of

X

that is compact and closed. If no such subset exists, explain why.

🔗

(e)

🔗

A

is a compact subset of

X,

must

A

be closed? Explain.

🔗

The message of Activity 17.2 is that compactness by itself is not related to closed or open sets. We will see later, though, that in some reasonable circumstances, compact sets and closed sets are related. For the moment, we take a short detour and ask if compactness is a topological property.

🔗