MA651 Topology. Lecture 8. Compactness 1.This text is based on the following books:• ”Topology” by James Dugundgji• ”Fundamental concepts of topology” by Peter O’Neil• ”Elements of Mathematics: General Topology” by Nicolas Bourbaki• ”Linear Algebra and Analysis” by Marc Zamansky• ”General topology I”, A.V. Arkhangel’skii and L.S. Pontryagin (Eds)I have intentionally made several mistakes in this text. The first homework assignment is to findthem.46 CompactnessCompactness is a topological concept which was originally inspired by properties of point setsin En. It was recognized quite early that certain kinds of sets had advantages over others incalculus. For example, a function continuous on a closed and bounded set achieves a maximumand a minimum.When topology was in its formative stages, there was no obvious was to generalize ”closed andbounded”. The prototype of the compactness notion is the following property of a segment(Lebesgue, 1903): any open covering of a segment [a, b] = {x ∈ R | a 6 x 6 b} of the real line Rcontains a finite subcovering. Later Borel generalized it for a finite dimensional Euclidean spaceEn, this theorem is learned in calculus as the Heine-Borel Theorem. A set A in Enis closed andbounded exactly when every cover of A by open sets can be reduced to a finite cover. That is, ifA ⊂[α∈Btα, where the sets tαare open, then it is possible to select finitely many of the tα’s suchthat A ⊂ tα1∪ tα2∪ ···∪tαn. This notion of reducibility of open covers generalizes to any space.A set with the property that each open cover has a finite reduction will be called compact.1The word ”compact” has an interesting history. Around 1906, Fr´echet used compact to mean thatevery infinite subset of A has a cluster point in A. This was probably motivated by the Boltzano-Weierstrass Theorem. Alexandroff and Urysohn, about 1924, used the notion of reduction of opencovers, but called it bicompactness. Bourbaki dropped the prefix ”bi”, but restricted ”himself”to Hausdorff spaces. When examples appeared in differential geometry of non-Hausdorff spaceshaving the reducibility property for open covers, Bourbaki labeled such spaces quasi-compact. Wewill use word compact for any arbitrary top ological space.Definition 46.1. An open cover of a topological space (X, T ) is a family of open sets Γ suchthat ∪Γ = X. If Ω is a subfamily of sets Ω ⊂ Γ and ∪Ω = X then Ω is called a subcover of thecover Γ.Definition 46.2. (Axiom of Borel-Lebesgue.) A topological space (X, T ) is said to be compact(or T -compact) if every open set cover Γ of X has a finite subcover µ.Example 46.1. Enis not compact. This is easiest to see when n = 1, where the open intervals]m − 1, m + 1[, as m takes on all integer values, form an open cover which cannot be reducedby as much as one set. A similar construction works in En. Take, for example, the spheresB%m((x1, . . . , xn),√m), where each coordinate xiof (x1, . . . , xn) is an integer.Example 46.2. Any indiscrete space is compact.Example 46.3. A discrete space is compact if and only if X is finite.Example 46.4. Let X = Z+, and let T consists of X, Ø and all sets [1, n] ∩Z+, where n ∈ Z+.Then, X is not T -compact.We can speak of a subset A of X as being T -compact or non T -compact by referring A to thesubspace topology induced by T .Definition 46.3. Let A ⊂ X, then A is T -c ompact if and only if A is TA-compact.In practice, compactness of A can always be decided without actually going into the relativetopology. We call a collection of T -open sets a cover of A if A is a subset of their union (but notnecessarily equal to their union). Compactness of A can then be tested by looking at T -c overs ofA, and TA-covers never have to be considered.Example 46.5. In Example (46.4), a subset A of Z+is compact exactly when it is boundedabove. That is, the only compact sets are the finite subsets of Z+. Conversely, each finite subsetis compact.Example 46.6. The empty set is a compact subset of any space, as is {x1, . . . xn} for any finitenumber of points x1, . . . , xnin X.2The next theorem is extremely useful and provide alternate formulations of the notion of com-pactness:Theorem 46.1. The following are equivalent:1. A topological space X is compact.2. Every family of closed subsets of X whose intersection is empty contains a finite subfamilywhose intersection is empty.3. Every filter on X has at least one cluster point.4. Every ultrafilter on X is convergent.Proof.(1) ⇒ (2) Assume (1). Let F be non-empty family of c losed subsets of X. Suppose that ∩F = Ø.We must produce a finite subset of F whose intersection is empty.Now, X = X −∩F = ∪{X −F | F ∈ F }. If F ∈ F , then X −F ∈ T , so {X −F | F ∈ F }if an open cover of X. Then, there are finitely many elements F1, . . . , Fnof F such thatX =n[i=1(X − Fi. Choose B = {F1, . . . , Fn}, and we then have B ⊂ F and ∩B = Ø.(2) ⇒ (1) Assume (2). Let Γ be an open cover of X. Then, Γ ∈ T and X = ∪Γ. Then, Ø = X −∪Γ =∩{X − γ | γ ∈ Γ}. If γ ∈ Γ, then X − γ is closed. By (2), there is a finite set of elementsγ1, . . . , γnof Γ such thatn\i=1(X − γi) = Ø. Then, X =n[i=1γi, and so X is a compact space.Proofs of the statements (3),(4) are left as a homework.Remarks:1. Sometimes it is said the the family of sets F has the finite intersection property if F satisfies(2).2. Note that, upon careful examination of the proof, the last theorem is just a restatement ofDeMorgan’s Laws as they relate to compactness.3. An ultrafilter on a set X is a filter F such that there is no filter which is strictly finer thanF .347 Compact subsets in RBefore considering compactness in general we consider compact subsets of E1.Theorem 47.1. Let A ⊂ E1. Then, A is compact if and only if A is closed and bounded.Proof. Suppose first that A is compact. Let {Ux} = {]x − 1, x + 1[| x ∈ A}. Then {Ux} is anopen cover of A. Since A is compact, there are finitely many points x1, . . . , xrin A such thatA ⊂r[i=1]xi−1, xi+ 1[. Then, A ⊂ [−η, η], where η = sup{|xi|+ 1 | 1 6 i 6 r}, hence is bounded.To show that A is closed, suppose that y ∈ A − A. In your homework derive a contradiction byshowing that {E1− [y −1j, y +1j] | j ∈ Z+} has no finite subset which covers A. Conversely,suppose that A is closed and bounded. We consider two cases.Case 1. A = [a, b] for some real numbers