Density topology and pointwise convergence

@ Applied General Topology c Universidad Polit´ecnica de Valencia Volume 4, No. 2, 2003 pp. 509–512 Density topology and pointwise convergence W ladys law Wilczy´ nski Dedicated to Professor S. Naimpally on the occasion of his 70th birthday. Abstract. We shall show that the space of all approximately continuous functions with the topology of pointwise convergence is not homeomorphic to its category analogue. 2000 AMS Classification: 54C35, 54C30, 26A15. Keywords: approximately continuous functions, I-approximately continuous functions, topology of pointwise convergence. 1. Density topology and pointwise convergence. Let Sbe a σ-algebra of Lebesgue measurable subsets of the real line R, L ⊂ S– a σ-ideal of null sets, B– a σ-algebra of subsets of Rposessing a property of Baire and I ⊂ B – a σ-ideal of sets of the first category. The sets of the first and the second category are considered only with respect to the natural topology. Recall that a point x0∈Ris a density point of the set A∈Sif and only if lim h→0+ λ(A∩[x0−h, x0+h]) 2h= 1, where λstands for the Lebesgue measure on S. Let Φ(A) be a set of all density points of A∈S. If we denote A∼Bin the case when A4B∈ L then we have (compare [6], Th. 22.2): Theorem 1.1. (1) Φ(A)∼Afor each A∈S(Lebesgue density theorem), (2) if A, B ∈Sand A∼B, then Φ(A) = Φ(B), (3) Φ(∅) = ∅,Φ(R) = R, (4) Φ(A∩B) = Φ(A)∩Φ(B)for each A, B ∈S. Observe that from 1 it follows immediately that Φ(A)∈Sfor each A∈S. The function Φ : S→Sis usually called a lower density operator. 510 W ladys law Wilczy´nski Theorem 1.2. ([6], Th. 22.5) A family Td={A∈S:A⊂Φ(A)}={Φ(E)\ P:E∈Sand P∈ L} is a topology on the real line stronger than the natural topology. The topology Tdis usually called the density topology. For further properties of Tdsee, for example, [4] or [9]. A real function of a real variable is called approximately continuous if it is continuous when the domain is equipped with the density topology and the range – with the natural topology. Since (R,Td) is a Tikhonov (completely regular) topological space ([4]), the density topology is the coarsest topology for the class of all approximately continuous functions. Observe that the following conditions are equivalent (see [8]) for a set A∈S: 1) 0 is a density point of A, 2) limn→∞ λ(A∩(−1 n,1 n)) 2 n = 1, 3) limn→∞ λ((n·A)∩(−1,1)) = 2 (where n·A={nx :x∈A}, 4) {χn·A∩(−1,1)}n∈Nconverges to χ(−1,1) in measure, 5) for each increasing sequence {nm}m∈Nof positive integers there exists a subsequence {nmp}p∈Nsuch that lim p→∞ χ(nmp·A)∩(−1,1) =χ(−1,1) almost everywhere. The equivalence of 1)-4) is immediate, while the equivalence of 4) and 5) follows from a well known theorem of Riesz. The above observation was a starting point to study a category analogue of a density point, density topology and approximate continuity. Definition 1.3. ([8]) We say that 0 is an I-density point of a set A∈ B if and only if for each increasing sequence {nm}m∈Nof positive integers there exists a subsequence {nmp}p∈Nsuch that lim p→∞ χ(nmp·A)∩(−1,1) =χ(−1,1) except on a set of the first category (in abbr. I-a.e.). We say that x0is an I-density point of A∈ B if and only if 0 is an I-density point of a set A−x0= ={x−x0:x∈A}. Let ΦI(A) be a set of all I-density points of A∈ B. If we denote now A∼B in the case when A4B∈ I, then we have Theorem 1.4. ([8]) (1) ΨI(A)∼Afor each A∈ B, (2) if A, B ∈ B and A∼B, then ΦI(A) = ΦI(B), (3) ΦI(∅) = ∅,ΦI(R) = R, (4) ΦI(A∩B) = ΦI(A)∩ΦI(B)for each A, B ∈ B. Theorem 1.5. ([8]) A family TI={A∈ B :A⊂ΦI(A)}={ΦI(E)\P:E∈ Band P∈ I} is a topology on the real line stronger than the natural topology. Density topology and pointwise convergence 511 The topology TIis called the I-density topology. For further properties of TIsee, for example, [8] or [3]. A real function of a real variable is called Iapproximately continuous if it is continuous when the domain is equipped with the I-density topology and the range – with the natural topology. Unfortunately, (R,TI) is a not a Tikhonov topological space ([8]). However, the coarsest topology for I-approximately continuous functions, which must be completely regular, is studied in details in [5] and [7]. We shall trace a description of such a topology (called the deep I-density topology) after [3]. Definition 1.6. A point x0∈Ris called a deep I-density point of A∈ B if there exists a closed (in the natural topology) set F⊂A∪ {x0}such that x0 is an I-density point of F. Let ΦID (A) be a set of all deep I-density points of A∈ B. Theorem 1.7. A family TID ={A∈ B :A⊂ΦID(A)}is a topology stronger than the natural topology and weaker than the I-density topology. Moreover TID is a completely regular topology. From the above theorem it follows immediately that the class of I-approximately continuous functions is equal to the class of deeply I-approximately continuous real functions of a real variable (i.e. the class of functions which are continuous when the domain is equipped with the deep I-density topology and the range with the natural topology). Since both spaces (R,Td) and (R,TID ) are Tikhonov topological spaces, it is reasonable to consider the spaces Cp(Rd) and Cp(RI) of all approximately continuous and all I-approximately continuous functions with the topology of pointwise convergence. The question: are Cp(Rd) and Cp(RI) homeomorphic seems to be interesting. In this note we shall try to find an answer. First of all, observe that the problem is not trivial by virtue of the following theorem: Theorem 1.8. The spaces (R,Td)and (R,TID)are not homeomorphic. Proof. Suppose that h:R→ onto Ris a homeomorphism between (R,Td) and (R,TID ). If E⊂Ris Td-connected set, then h(E) is TID -connected set. From [4] it follows that the family of all Td-connected sets coincides with the family of all sets connected in the natural topology (i.e. with the family of all intervals — open, half-open, closed, bounded or unbounded). The same holds for the topology TI(see [8]). Since TDI is between TIand the natural topology, it has the same family of connected sets. So for hwe see that the image of an arbitrary interval is an interval. From this it is easy to conclude that his a strictly monotone and continuous (in the sense that both the domain and the range are equipped with the natural topology) function, in fact his a homeomorphism from (R,nat) to (R,nat). Let E∈ Tdbe a set which is nowhere dense in the natural topology (for example E=C∩Φ(C), where Cis a nowhere dense Cantor set of positive measure). Then h(E) is also nowhere dense in the natural topology. But from the definition of deep I-density point 512 W ladys law Wilczy´nski and deep I-density topology it follows immediately that each set in TID is of the second category in fact, it must contain a nondegenerate closed interval (see the proof of Th. 2 below). So h(E)/∈ TID – a contradiction.  Theorem 1.9. The spaces Cp(Rd)and Cp(RI)are not homeomorphic. Proof. Suppose that Cp(Rd) and Cp(RI) are homeomorphic. Then d(R,Td) = d(R,TID ) ([1], p. 26), where ddenotes the density of the topological space, i.e. the smallest cardinal number of dense subsets of this space. But from the definition of deep I-density topology it follows that each TID -open set includes a (nondegenerate) closed interval, because TID-open set includes a closed set (in the natural topology) of the second category. Hence the set E⊂Ris dense in the topology TID if and only if it is dense in the natural topology and d(R,TDI ) = ℵ0. Simultaneously if the set Eis dense in the topology Td, then λ∗(E)>0 (in fact, λ∗(E∩(a, b)) = b−afor each interval (a, b)). So d(R,Td)>ℵ0– a contradiction. Observe that the exact value of d(R,Td) depends essentially of the system of axioms (see [2]).  References [1] A.V. Arkhangel’skiˇı, Prostranstwa funktsiˇı v topologii potochechnoˇı skhodimosti (Spaces of functions in the topology of pointwise convergence), General topology, spaces of functions and dimension, MGU Moskva 1985 (in Russian). [2] T. Bartoszy´nski, H. Judah, Set theory. On the structure of the Real Line, A.K. Peters, Wellesley 1995. [3] K. Ciesielski, L. Larson, K. Ostaszewski, I-density continuous functions, Memoirs of AMS 515(1994). [4] C. Goffman, C.J. Neugebauer, T. Nishiura, Density topology and approximate continuity, Duke Math. J. 28 ( 1961), 497–505. [5] E. Lazarow, The coarsest topology of I-approximately continuous functions, Comm. Math. Univ. Carolinae 27 (1986), 695–704. [6] J.C. Oxtoby Measure and category, Springer Verlag, New York-Heidelberg-Berlin, (1980). [7] W. Poreda, E. Wagner-Bojakowska, The topology of I-approximately continuous functions, Radovi Mat. 2(1986), 263–277. [8] W. Poreda, E. Wagner-Bojakowska, W. Wilczy´nski, A category analogue of the density topology Fund. Math. 125 (1985), 167–173. [9] F.D. Tall, The density topology Pacific J. Math. 62 (1976), 275–284. Received January 2002 Revised September 2002 W ladys law Wilczy´ nski Faculty of Mathematics, Chair of Real Functions, L´od´z University, Stefana Banacha 22, 90-238 L´od´z, Poland E-mail address:[email protected]