A new approach to the Katětov-Tong theorem
aa r X i v : . [ m a t h . GN ] J a n A NEW APPROACH TO THE KAT ˇETOV-TONG THEOREM
G. BEZHANISHVILI, P. J. MORANDI, AND B. OLBERDING
Abstract.
We give a new proof of the Katˇetov-Tong theorem. Our strategy is to first provethe theorem for compact Hausdorff spaces, and then extend it to all normal spaces. The keyingredient is how the ring of bounded continuous real-valued functions embeds in the ringof all bounded real-valued functions. In the compact case this embedding can be describedby an appropriate statement, which we prove implies both the Katˇetov-Tong theorem anda version of the Stone-Weierstrass theorem. We then extend the Katˇetov-Tong theorem toall normal spaces by showing how to extend upper and lower semicontinuous real-valuedfunctions to the Stone- ˇCech compactification so that the less than or equal relation betweenthe functions is preserved. Introduction
For a topological space X let B ( X ) be the ring of all bounded real-valued functions and C ∗ ( X ) the subring consisting of continuous functions. We recall (see, e.g., [5, Def. 1, p. 360])that f ∈ B ( X ) is upper semicontinuous if f − ( −∞ , λ ) is open for all λ ∈ R and f is lower semicontinuous if f − ( λ, ∞ ) is open for all λ ∈ R . It is well known (see, e.g., [5,Prop. 3, p. 363]) that f is upper semicontinous iff f ( x ) = inf U ∈N x sup y ∈ U f ( y ) and f is lowersemicontinuous iff f ( x ) = sup U ∈N x inf y ∈ U f ( y ), where N x is the set of all open neighborhoodsof x . Let USC ( X ) = { f ∈ B ( X ) | f is upper semicontinuous } LSC ( X ) = { f ∈ B ( X ) | f is lower semicontinuous } . We can then formulate the famous Katˇetov-Tong theorem as follows:
Katˇetov-Tong Theorem (KT):
Let X be a normal space. If f ∈ USC ( X ) and g ∈ LSC ( X )with f ≤ g , then there is h ∈ C ∗ ( X ) such that f ≤ h ≤ g .Neither Katˇetov’s proof [10, 11] nor Tong’s [12] simplifies in the compact setting. Wegive a different proof of (KT) by first proving it for compact Hausdorff spaces. Our proofis based on [1] where we gave a necessary and sufficient condition for a completely regularspace X to be compact in terms of how C ∗ ( X ) embeds in B ( X ). We also use Dilworth’scharacterization of upper and lower semicontinuous functions [7, Lem. 4.1].To obtain the full version of (KT) for an arbitrary normal space X , we use the Stone-ˇCech compactification βX of X . The key observation in this part of the proof is that if f ∈ USC ( X ), g ∈ LSC ( X ), and f ≤ g , then we can extend f to F ∈ USC ( βX ) and g to Mathematics Subject Classification.
Key words and phrases.
Normal space; compact Hausdorff space; continuous real-valued function; upperand lower semicontinuous functions; Stone- ˇCech compactification; ℓ -algebra. G ∈ LSC ( βX ) so that F ≤ G . This allows us to use the already established (KT) for compactHausdorff spaces to produce a continuous function between F and G , whose restriction to X is then the desired continuous function on X .We conclude the article by showing that a version of the Stone-Weierstrass theorem alsofollows from our approach. In order to formulate the Stone-Weierstrass theorem, we pointout that B ( X ) is not only a ring, but an R -algebra and C ∗ ( X ) is an R -subalgebra of B ( X ).We recall that the uniform norm is defined on B ( X ) by k f k = sup f ( X ). We then have ametric space structure on B ( X ), where the distance between f, g is k f − g k . Elementaryanalysis arguments show that B ( X ) and C ∗ ( X ) are complete as metric spaces with respectto the uniform norm. If X is compact Hausdorff, then C ∗ ( X ) coincides with the R -algebra C ( X ) of all continuous real-valued functions on X . Stone-Weierstrass Theorem (SW): If X is compact Hausdorff and A is an R -subalgebraof C ( X ) which separates points of C ( X ), then A is uniformly dense in C ( X ).In addition to B ( X ) being an R -algebra, there is a natural order ≤ on B ( X ) defined by f ≤ g iff f ( x ) ≤ g ( x ) for all x ∈ X . It is elementary to see that the following conditionshold on B ( X ):(1) B ( X ) is a lattice; (2) f ≤ g implies f + h ≤ g + h ;(3) 0 ≤ f, g implies 0 ≤ f g ;(4) 0 ≤ f and 0 ≤ λ ∈ R imply 0 ≤ λf .Thus, B ( X ) is a lattice-ordered algebra or ℓ -algebra for short, and C ∗ ( X ) is an ℓ -subalgebraof B ( X ), where we recall that an ℓ -subalgebra is an R -subalgebra which is also a sublattice.We can replace the R -subalgebra condition in (SW) with an ℓ -subalgebra condition andarrive at the following version of the Stone-Weierstrass theorem. Stone-Weierstrass for ℓ -subalgebras (SW ℓ ): If X is compact Hausdorff and A is an ℓ -subalgebra of C ( X ) which separates points of C ( X ), then A is uniformly dense in C ( X ).We conclude the article by showing how to derive this version of the Stone-Weierstrasstheorem from our approach.2. The Katˇetov-Tong Theorem
Let X be a completely regular space. In [1] we showed that X is compact iff the inclusion C ∗ ( X ) ⊆ B ( X ) satisfies Condition (C) below. This will play an important role in proving(KT) for compact Hausdorff spaces. To formulate (C), we point out that if S ⊆ B ( X ) isbounded, then the least upper bound W S and the greatest lower bound V S exist in B ( X )and are pointwise. Definition 2.1.
Let X be completely regular, S, T ⊆ C ∗ ( X ) bounded, and 0 < ε ∈ R .(C) If V S + ε ≤ W T in B ( X ), then there are finite S ⊆ S and T ⊆ T with V S ≤ W T . That is, f ∨ g, f ∧ g exist in B ( X ). They are defined by ( f ∨ g )( x ) = max { f ( x ) , g ( x ) } and ( f ∧ g )( x ) =min { f ( x ) , g ( x ) } for each x ∈ X . NEW APPROACH TO THE KATˇETOV-TONG THEOREM 3
Remark 2.2. (1) The presence of ε in (C) is necessary. To see this, identify R with a subalgebra of B ( X ), and let S = { η ∈ R | < η } and T = { λ ∈ R | λ < } . Then V S ≤ W T butthere are no finite S ⊆ S and T ⊆ T satisfying V S ≤ W T .(2) In [1] we considered Condition (C) for more general embeddings A → B ( X ). For thepurposes of this paper we concentrate on the inclusion C ∗ ( X ) ⊆ B ( X ). Theorem 2.3. ([1, Thm. 2.6(2)])
Let X be completely regular. Then X is compact iff theinclusion C ∗ ( X ) ⊆ B ( X ) satisfies (C) . Remark 2.4.
Our proof of (KT) for compact Hausdorff spaces (see Lemma 2.6) only needsone implication of Theorem 2.3, that the inclusion C ∗ ( X ) ⊆ B ( X ) satisfies (C) if X iscompact.The following result uses Dilworth’s lemma [7, Lem. 4.1] characterizing upper and lowersemicontinuous functions: Let X be a completely regular space and f ∈ B ( X ). Then f ∈ USC ( X ) iff f is a pointwise meet of continous functions, and f ∈ LSC ( X ) iff f is apointwise join of continous functions. Lemma 2.5.
Let X be compact Hausdorff, f ∈ USC ( X ) , g ∈ LSC ( X ) , and ε > . If f + ε ≤ g , then there is a ∈ C ( X ) with f ≤ a ≤ g .Proof. Since X is compact Hausdorff, it follows from Urysohn’s lemma that X is completelyregular. Therefore, by Dilworth’s lemma, f = V S and g = W T for some S, T ⊆ C ( X ).Thus, by Theorem 2.3, there exist finite S ⊆ S and T ⊆ T with f ≤ V S ≤ W T ≤ g . Set a = V S . Then a ∈ C ( X ) and f ≤ a ≤ g . (cid:3) We are ready to prove (KT) in the compact Hausdorff setting. For this we utilize atechnique that goes back to Dieudonn´e [6], and was used by Edwards [8, p. 21] and Blatterand Seever [4, pp. 32-33].
Lemma 2.6.
Let X be compact Hausdorff. If f ∈ USC ( X ) and g ∈ LSC ( X ) with f ≤ g ,then there is a ∈ C ( X ) such that f ≤ a ≤ g .Proof. Let f ∈ USC ( X ), g ∈ LSC ( X ), and f ≤ g . By induction we construct a sequence { a n | n ≥ } in C ( X ) such that for each n ≥ f − / n ≤ a n ≤ g (1) a n − − / n − ≤ a n ≤ a n − + 1 / n − . (2)For the base case, since ( f − /
2) + 1 / ≤ g , by Lemma 2.5 there is a ∈ C ( X ) with f − / ≤ a ≤ g . Set a = a . Then (1) and (2) are satisfied for n = 1. Suppose that m ≥ a , . . . , a m ∈ C ( X ) satisfying (1) and (2) for all 1 ≤ n ≤ m . By (1) for n = m we get f ≤ a m + 1 / m . In addition, it is clear that a m − / m +1 ≤ a m + 1 / m . Thus, f ∨ ( a m − / m +1 ) ≤ a m + 1 / m . Since f, a m ≤ g , it is also clear that f ∨ ( a m − / m +1 ) ≤ g . So f ∨ ( a m − / m +1 ) ≤ g ∧ ( a m + 1 / m ) . G. BEZHANISHVILI, P. J. MORANDI, AND B. OLBERDING
Since ( a ∨ b ) + c = ( a + c ) ∨ ( b + c ) holds in B ( X ), (cid:2) ( f − / m +1 ) ∨ ( a m − / m ) (cid:3) + 1 / m +1 =( f − / m +1 + 1 / m +1 ) ∨ ( a m − / m + 1 / m +1 ) = f ∨ ( a m − / m +1 ) . Consequently, (cid:2) ( f − / m +1 ) ∨ ( a m − / m ) (cid:3) + 1 / m +1 ≤ g ∧ ( a m + 1 / m ) . By Lemma 2.5, there is a m +1 ∈ C ( X ) satisfying( f − / m +1 ) ∨ ( a m − / m ) ≤ a m +1 ≤ g ∧ ( a m + 1 / m ) . Therefore, f − / m +1 ≤ a m +1 ≤ ga m − / m ≤ a m +1 ≤ a m + 1 / m . Thus, (1) and (2) hold for n = m + 1. By induction we have produced the desired sequence.Equation (2) implies that { a n } is a Cauchy sequence, so has a uniform limit a ∈ C ( X ). Foreach x ∈ X , (1) yields f ( x ) − / n ≤ a n ( x ) ≤ g ( x ) for each n . Taking limits as n → ∞ gives f ( x ) ≤ a ( x ) ≤ g ( x ). Therefore, f ≤ a ≤ g . (cid:3) To extend (KT) to an arbitrary normal space we require the following lemma.
Lemma 2.7. ([3, Lem 7.2])
Let X be a dense subspace of a compact Hausdorff space Y . (1) If f ∈ USC ( X ) , define U ( f ) on Y by U ( f )( y ) = inf U ∈N y sup x ∈ U ∩ X f ( x ) . Then U ( f ) ∈ USC ( Y ) and extends f . (2) If f ∈ LSC ( X ) , define L ( f ) on Y by L ( f )( y ) = sup U ∈N y inf x ∈ U ∩ X f ( x ) . Then L ( f ) ∈ LSC ( Y ) and extends f . Remark 2.8.
There can exist upper semicontinuous extensions of f ∈ USC ( X ) to Y otherthan U ( f ). For example, let X be an infinite discrete space and Y an arbitrary compactifi-cation of X . If f = 0 on X , then U ( f ) = 0 on Y , while the characteristic function of Y \ X is another upper semicontinuous extension of f because Y \ X is closed. Similarly, there canexist lower semicontinuous extensions of f ∈ LSC ( X ) to Y other than L ( f ).We are ready to prove (KT) for an arbitrary normal space X . Let βX be the Stone- ˇCechcompactification of X . Without loss of generality we may assume that X is a subspace of βX . Since X is normal, if C, D are disjoint closed subsets of X , it is a simple consequenceof Urysohn’s lemma that cl βX ( C ) ∩ cl βX ( D ) = ∅ (see, e.g., [9, Cor. 3.6.4]). Theorem 2.9. (Katˇetov-Tong)
Let X be a normal space. If f ∈ USC ( X ) and g ∈ LSC ( X ) with f ≤ g , then there is h ∈ C ∗ ( X ) such that f ≤ h ≤ g . NEW APPROACH TO THE KATˇETOV-TONG THEOREM 5
Proof.
Set F = U ( f ) and G = L ( g ). By Lemma 2.7, F ∈ USC ( βX ) and extends f ,and G ∈ LSC ( βX ) and extends g . We show F ≤ G . If not, there are y ∈ βX and λ, η ∈ R with F ( y ) > η > λ > G ( y ). Let U be an open neighborhood of y . Since F ( y ) = inf U ∈N y sup x ∈ U ∩ X f ( x ), we have sup x ∈ U ∩ X f ( x ) > η for each open U ∈ N y . Therefore, U ∩ f − [ η, ∞ ) = ∅ . Thus, y ∈ cl βX f − [ η, ∞ ). Similarly, y ∈ cl βX g − ( −∞ , λ ]. Because f isupper semicontinuous, f − [ η, ∞ ) is closed and since g is lower semicontinuous, g − ( −∞ , λ ]is closed. As X is normal, f − [ η, ∞ ) ∩ g − ( −∞ , λ ] = ∅ since their closures in βX are notdisjoint. This is a contradiction to f ≤ g . Therefore, F ≤ G . By Lemma 2.6, there is a ∈ C ( βX ) with F ≤ a ≤ G . If h is the restriction of a , then h ∈ C ∗ ( X ) and f ≤ h ≤ g . (cid:3) Remark 2.10.
The key step in the proof of Theorem 2.9 is to show that if f ∈ USC ( X )and g ∈ LSC ( X ) with f ≤ g , then U ( f ) ≤ L ( g ).(1) If X is not normal, it need not be true that f ≤ g implies U ( f ) ≤ L ( g ). To seethis, let X be a completely regular but not normal space. Then there are disjointclosed sets C, D of X with cl βX ( C ), cl βX ( D ) having nonempty intersection. Let y ∈ cl βX ( C ) ∩ cl βX ( D ), f be the characteristic function of C , and g the characteristicfunction of X \ D . It is easy to see that f ∈ USC ( X ) and g ∈ LSC ( X ). Since C and D are disjoint, f ≤ g . Let U be an open neighborhood of y . Then U ∩ C is nonempty,so sup x ∈ U ∩ X f ( x ) = 1. Therefore, U ( f )( y ) = 1. Similarly, U ∩ D is nonempty, soinf x ∈ U ∩ X g ( x ) = 0, and hence L ( g )( y ) = 0. Thus, U ( f ) L ( g ).(2) We cannot replace βX with an arbitrary compactification of X . For example, let X be an infinite discrete space and Y the one-point compactification of X . Let A be aninfinite subset of X whose complement is also infinite, f the characteristic functionof A , and g = f . Trivially f ∈ USC ( X ), g ∈ LSC ( X ), and the same argument asabove shows that U ( f )( ∞ ) = 1 and L ( g )( ∞ ) = 0. Thus, U ( f ) L ( g ).3. The Stone-Weierstrass Theorem for ℓ -subalgebras In this final section we show how to derive (SW ℓ ) from (C). Definition 3.1.
Let X be completely regular and let A be an ℓ -subalgebra of B ( X ).(1) Call f ∈ B ( X ) closed relative to A if there is S ⊆ A with f = V S .(2) Call g ∈ B ( X ) open relative to A if there is T ⊆ A with g = W T .(3) Call h ∈ B ( X ) clopen relative to A if h is both closed and open relative to A . Remark 3.2. (1) It is easy to see that if S is a subset of X , then the characteristic function χ S is closedrelative to C ∗ ( X ) iff S is a closed subset of X , and χ S is open relative to C ∗ ( X ) iff S is open in X . This motivates the terminology of Definition 3.1.(2) It is also easy to see that if f ∈ B ( X ) is open (resp. closed) relative to A , then so is f + λ . Lemma 3.3.
Let X be compact Hausdorff. The clopen elements of B ( X ) relative to A arein the uniform closure of A in B ( X ) . G. BEZHANISHVILI, P. J. MORANDI, AND B. OLBERDING
Proof.
Let h be clopen in B ( X ) relative to A and let ε >
0. Then ( h + ε/
2) + ε/ ≤ h + ε .By Remark 3.2(2), h + ε/ h + ε is open relative to A . Therefore, h + ε/ V S and h + ε = W T for some S, T ⊆ A . Since X is compact Hausdorff, by Theorem 2.3, theinclusion C ( X ) ⊆ B ( X ) satisfies Condition (C). Because A ⊆ C ( X ), it follows from (C)that there are finite S ⊆ S and T ⊆ T with h + ε/ ≤ V S ≤ W T ≤ h + ε . Let a = V S .Then h + ε/ ≤ a ≤ h + ε , and a ∈ A since A is an ℓ -subalgebra of C ( X ). Thus, k a − h k ≤ ε .Since ε is arbitrary, this shows that h is in the uniform closure of A . (cid:3) Remark 3.4.
While we do not need it, the converse of Lemma 3.3 that each element in theuniform closure of A is clopen relative to A is also true (see, e.g., [2, Lem 3.16(2)]).The last ingredient needed for (SW ℓ ) is the following lemma, the first two items of whichare in [1, Lem 2.8]) and the third is an easy consequence of the first two. Lemma 3.5.
Let X be compact Hausdorff and let A be an ℓ -subalgebra of C ( X ) whichseparates points of X . (1) If f ∈ USC ( X ) , then f is closed relative to A . (2) If g ∈ LSC ( X ) , then g is open relative to A . (3) If h ∈ C ( X ) , then h is clopen relative to A . Remark 3.6.
While we do not need it, the converse statements to the statements inLemma 3.5 are true, and are easy to prove (see, e.g., [5, Thm. 4, p. 362]).We are ready to prove (SW ℓ ). Theorem 3.7. If X is compact Hausdorff and A is an ℓ -subalgebra of C ( X ) which separatespoints of C ( X ) , then A is uniformly dense in C ( X ) .Proof. By Lemma 3.5, elements of C ( X ) are clopen relative to A , so lie in the uniform closureof A in B ( X ) by Lemma 3.3. Thus, A is dense in C ( X ). (cid:3) Remark 3.8.
While we have assumed that A is an ℓ -subalgebra of B ( X ), it is sufficient toassume that A is simply a vector sublattice of B ( X ). Indeed, the existence of multiplicationon A is not needed in the proofs. References [1] G. Bezhanishvili, P. J. Morandi, and B. Olberding,
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