Compactness · Definition

Compact space

topologycompactnesstopological spaces

Before you start you need to know

It is the property that carries the finiteness of closed bounded intervals over to general spaces.

Definition

A topological space \(X\) is compact if every open cover \(\mathcal{U}\) of \(X\) has a finite subcover.

\[X=\bigcup_{U\in\mathcal{U}}U\ \Rightarrow\ X=U_1\cup\dots\cup U_n,\ U_i\in\mathcal{U}\]

Compact does not mean small or finite; it means one cannot escape to infinity: however many open sets are used to cover the space, finitely many always suffice. Every finite space is compact, and so is the cofinite topology on any set, even an infinite one.

In \(\mathbb{R}\), the Heine-Borel theorem says the compact sets are exactly the closed bounded ones. That is why \([0,1]\) is compact while \((0,1)\) is not (the cover \(\{(1/n,1)\}\) cannot be thinned) and neither is \(\mathbb{R}\) (\(\{(-n,n)\}\) cannot be thinned).

A subset \(A\subseteq X\) is compact as a subspace if and only if every family of open sets of \(X\) covering it contains a finite subfamily that still covers it. The union of two compact sets is compact; with infinitely many it can fail, and a closed subset of a compact set is again compact.

To show a space is not compact one failing cover is enough; to show it is, one must handle an arbitrary cover. The next two concepts describe how compactness interacts with closed sets and continuous maps.

Think about it

Is \(\mathbb{Q}\cap[0,1]\) compact with the usual topology?

No: it is not closed in \(\mathbb{R}\). Open intervals around each rational that avoid \(\sqrt2/2\), with radii shrinking toward it, form a cover with no finite subcover.

Why does being infinite not prevent compactness?

Because the definition asks for a finite subcover, not finitely many points; \([0,1]\) and the cofinite space are compact and infinite.

Exercises 10

  1. Which of these subsets of \(\mathbb{R}\) is compact?
  2. Is a finite topological space compact?
  3. For integers \(n\ge 1\), the intervals \((-n,n)\) cover \(\mathbb{R}\). What is the smallest \(n\) with \([-5,5]\subseteq(-n,n)\)?
  4. Let \(X\) be an infinite set with the discrete topology. Is it compact?
  5. Which subset of \(\mathbb{R}\) is compact?
  6. Let \(X=\{a,b,c,d,e\}\) with the discrete topology and the cover \(\{a,b\},\{c,d\},\{e\},\{b,c\},\{d,e\},\{a,e\}\). What is the least number of sets in a subcover?
  7. Let \(X\) be an infinite set with the cofinite topology (the open sets are \(\varnothing\) and complements of finite sets). Is it compact?
  8. To prove that \(A=(0,1)\) is not compact it suffices to exhibit:
  9. Let \(K_1,K_2\) be compact subsets of a space \(X\). Which statement is true?
  10. Let \(X=\mathbb{N}\) with open sets \(\varnothing\) and \(U_k=\{k,k+1,\dots\}\), \(k\ge1\). Is it compact?

References

  • James R. Munkres, Topology, 2nd edition, Pearson, 2000. §26
  • Stephen Willard, General Topology, Dover reprint, Dover Publications, 2004. cap. 5