19.2. THE ALEXANDER SUB-BASIS THEOREM 507

The complement is taken with respect to X̃ and so the open sets, KC are basic open setswhich contain ∞.

The reason this is called a compactification is contained in the next lemma.

Lemma 19.1.27 If (X ,τ) is a locally compact Hausdorff space, then(

X̃ , τ̃)

is a com-

pact Hausdorff space. Also if U is an open set of τ̃, then U \{∞} is an open set of τ .

Proof: Since (X ,τ) is a locally compact Hausdorff space, it follows(

X̃ , τ̃)

is a Haus-dorff topological space. The only case which needs checking is the one of p ∈ X and ∞.Since (X ,τ) is locally compact, there exists an open set of τ, U having compact closurewhich contains p. Then p ∈U and ∞ ∈UC and these are disjoint open sets containing thepoints, p and ∞ respectively. Now let C be an open cover of X̃ with sets from τ̃ . Then ∞

must be in some set, U∞ from C , which must contain a set of the form KC where K is acompact subset of X . Then there exist sets from C , U1, · · · ,Ur which cover K. Therefore,a finite subcover of X̃ is U1, · · · ,Ur,U∞.

To see the last claim, suppose U contains ∞ since otherwise there is nothing to show.Notice that if C is a compact set, then X \C is an open set. Therefore, if x ∈U \{∞} , andif X̃ \C is a basic open set contained in U containing ∞, then if x is in this basic open setof X̃ , it is also in the open set X \C ⊆U \{∞} . If x is not in any basic open set of the formX̃ \C then x is contained in an open set of τ which is contained in U \{∞}. Thus U \{∞}is indeed open in τ . ■

Lemma 19.1.28 Let (X ,τ) be a topological space and let B be a basis for τ . Then Kis compact if and only if every open cover of basic open sets admits a finite subcover.

Proof: Suppose first that X is compact. Then if C is an open cover consisting of basicopen sets, it follows it admits a finite subcover because these are open sets in C .

Next suppose that every basic open cover admits a finite subcover and let C be an opencover of X . Then define C̃ to be the collection of basic open sets which are contained insome set of C . It follows C̃ is a basic open cover of X and so it admits a finite subcover,{

U1, · · · ,Up}

. Now each Ui is contained in an open set of C . Let Oi be a set of C whichcontains Ui. Then

{O1, · · · ,Op

}is an open cover of X . ■

Actually, there is a profound generalization of this lemma.

19.2 The Alexander Sub-basis TheoremThe Hausdorff maximal theorem is one of several convenient versions of the axiom ofchoice. For a discussion of this, see the appendix on the subject. There is this one, the wellordering principal, and Zorn’s lemma. They are all equivalent to the axiom of choice andwhich one you use is a matter of taste.

Theorem 19.2.1 (Hausdorff maximal principle) Let F be a nonempty partiallyordered set. Then there exists a maximal chain.

The main tool in the study of products of compact topological spaces is the Alexandersubbasis theorem which is presented next. Recall a set is compact if every basic open coveradmits a finite subcover, Lemma 19.1.28. This was pretty easy to prove. However, there isa much smaller set of open sets called a subbasis which has this property. The proof of thisresult is much harder.