A category analogue of the density topology non-homeomorphic with the I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {I}$$\end{document}-density topology

The paper deals with the category analogue of a density point and a density topology (with respect to a Lebesgue measure) on the real line which is different from the I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {I}$$\end{document}-density topology considered in Poreda et al. (Fundam Math 125:167–173, 1985; Comment Math Univ Carol 26:553–563, 1985). This topology called the intensity topology, manifests several properties analogous to that of I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {I}$$\end{document}-density topology, but there are also differences. The class of function which are continuous as functions from R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document} equipped with an intensity topology to R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document} equipped with the natural topology is included in the first class of Baire Darboux functions.


Introduction
In [8] one can find a characterization of a Lebesgue density point which uses only the σ -ideal of nullsets. This was a starting point for the definition of category density point, called the I-density point and for the construction of I-density topology, which was studied later in numerous papers. In this paper we shall formulate another characterization of a Lebesgue density point, again not using a Lebesgue measure but only the σ -ideal of nullsets. This characterization leads again to a definition of category density point, which is, however, not equivalent to I-density and to a category density topology, which is different from the I-density topology.
In the sequel L will denote the σ -algebra of Lebesgue measurable sets on the real line, N -the σ -ideal of nullsets, B-the σ -algebra of sets having the Baire property, Wladyslaw.Wilczynski@wmii.uni.lodz.pl 1 Faculty of Mathematics and Computer Science, University of Łódź, 90-238 Lodz, Poland I-the σ -ideal of sets of the first category, λ will stand for the Lebesgue measure. We shall use also the following denotation: A + x = {t + x : t ∈ A} and x A = {xt : t ∈ A}.
Recall that x 0 ∈ R is a density point of A ∈ L if and only if From (1) it follows immediately that : L → L (in fact : L → G δσ ). The family T d = {A ∈ L : A ⊂ (A)} is a topology on R (called the density topology), which is essentially stronger than a natural topology.
Since x 0 is a density (resp. dispersion) point of A ∈ L if and only if it is simultaneously a left-hand and a right-hand density (resp. dispersion) point of A, we shall deal in the sequel with the right-hand case, the remaining being obvious. The expression "in measure" will always mean with respect to the Lebesgue measure restricted to [0, 1], [−1, 0] or [−1, 1].
In [8] it was observed that x 0 is a right-hand density (resp. dispersion) point of A ∈ L if and only if the sequence {χ (n·(A−x 0 )∩[0,1]) } n∈N of characteristic functions converges in measure to χ [0,1] (resp. to χ ∅ ). By virtue of theorem of Riesz, x 0 is a right-hand density (resp. dispersion) point of A ∈ L if and only if for each increasing sequence {n m } m∈N of positive integers there exists a subsequence {n m p } p∈N such that {χ n m p · (A − x 0 ) ∩ [0, 1]} p∈N converges to χ [0,1] (resp. to χ ∅ ) almost everywhere. The last condition indeed does not use the Lebesgue measure, only the σ -ideal N is important.
The convergence in category means that for each increasing sequence {n m } m∈N of positive integers there exists a subsequence {n m p } p∈N such that {χ n m p ·(A−x 0 )∩[ −1,1] } p∈N converges to χ [−1,1] I-almost everywhere (i.e. except on a set belonging to I) (compare [13]). I-dispersion points, right-hand I-density or I-dispersion points and left-hand Idensity or I-dispersion points are defined in an obvious way. If I (A) = {x ∈ R : x is an I-density point of A} for A ∈ B, then the operator I has all properties analogous to properties (1)-(4) of , also I : B → B. The family T I = {A ∈ B : A ⊂ I (A)} is a topology on R (called the I-density topology), which is essentially stronger than a natural topology (see [8] or [3]).

New characterization
Suppose that A is a Lebesgue measurable set. Put a n = λ(A ∩ [ 1 n+1 , 1 n ]) for n ∈ N and let b n = n(n + 1)a n be an average density of the set A on the interval [ 1 n+1 , 1 n ]. Definition 2 We shall say that 0 is the right-hand C-density point of A (C-dispersion point of A) if and only if lim n→∞ The definition of the left-hand C-density point(C-dispersion point) is formulated in an obvious way. 0 is the C-density (C-dispersion) point of A if and only if it is simultaneously the right-hand and the left-hand C-density (C-dispersion) point of A. We say that x 0 is the right-hand (left-hand) Observe that 0 is the right-hand C-density point of A if and only if it is the right-hand C-dispersion point of A .

Theorem 3 0 is the right-hand C-dispersion point of A if and only if 0 is the right-hand dispersion point of A.
Proof "⇒" Take ε > 0. There exists n 0 ∈ N such that for n ≥ n 0 we have 1 This implies lim n→∞ 1 n = 0 and 0 is the right-hand dispersion point of A.
"⇐" Take ε > 0. Fix a positive integer k > 3 ε +1. Then we have n nk−1 < ε 3 for each n ∈ N. Since lim n→∞ At last for n ≥ n 0 and m between nk − 1 and (n + 1)k − 1 we have 1 which means that 0 is the right-hand C-dispersion point of A.

Remark 4
Obviously similar theorem holds also for the left-hand C-dispersion point of A. Also 0 is the right-hand (or left-hand) C-density point of A if and only if it is the right-hand (or left-hand) density point of A.

Intensity topology
Now we shall formulate the definition of C-dispersion (and C-density) point without using the Lebesgue measure. By virtue of Theorem 3 it will be another characterization of the dispersion (and density) point. This will enable us to introduce the category analogue of the density and dispersion point.
Put In the sequel we shall use the following denotation: If x = 0 we shall write A n rather than A n (0).
Theorem 6 i is a lower density operator, i.e. Proof Exactly the same as in [8] for T I .

Proof
We shall show that the notion of intensity point is not equivalent to that of I-density point. However, the topology T i shares most of the properties (but not all) with the topology T I .
At the same time the set A∩[ 1 n+1 , 1 n ] includes a non-empty open set for each n ∈ N, from which it follows easily that 0 is not a right-hand I-dispersion point of A.

Theorem 9
There exists a set A ⊂ [0, 1] such that 0 is the right-hand I-dispersion point of A but not the right-hand rarefaction point of A.
. Let I 1 1 and I 2 1 be closed components of C 1 numbered from the left to the right. Suppose that we have defined C 1 , C 2 , . . . , C n−1 such that C n−1 ⊂ [ 1 6 n−1 , 1 6 n−2 ] and I 1 n−1 , . . . , I 2 n−1 n−1 is the sequence of closed components of C n−1 numbered from the left to right. We shall define C n ⊂ [ 1 6 n , 1 6 n−1 ] in the following way: where m is a positive integer such that the center of 1 6 Then C n is the union of 2 n closed intervals I 1 n , . . . , 7 10 for each n ∈ N and t ∈ [0, 1] and 0 is not a right-hand rarefaction point of A. Indeed observe that 1 5·6 n−1 (6 n−1 + 2 · 6 n−2 + · · · + 2 n−2 · 6 + 2 n−1 ) = 1 5 (1 + 2 6 + 2 2 6 2 + · · · + 2 n−1 6 n−1 ) < Since this is a perfect nowhere dense set, the proof is finished. Now we shall present some basic properties of the topology T i . The proofs go exactly in the same way as for the topology T I , so we shall give only references at the end of the following theorem.

Theorem 10
The topology T i on R has the following properties:

a subset A of R is closed and discrete with respect to T i if and only if
For the proofs of 1, 3, 5 see [3], p. 37, proof of 2, 6 see [9], proof of [4] see [14] and of 7 see [4].
The following property of T i needs a separate proof.
Similarly as above we can find a point x 0 and d n → n→∞ x 0 . Observe also that the sequence {c 2n−1 } n∈N is increasing and {c 2n } n∈N is decreasing. From the construction it follows that d n − c n ≥ dist(x 0 , [c n , d n ]). Consider the sequence {[c 2n , d 2n ]} n∈N . Let k n and m n for n ∈ N be positive integers such that m n i=k n Both sequences {k n } n∈N , {m n } n∈N are increasing and tend to infinity. Moreover lim inf n→∞ m n k n ≥ 2. Then lim sup n→∞

Intense continuity
To introduce the notion of intense continuity (right-or left intense continuity we shall need the notion of interior intensity point of an arbitrary set A ⊂ R. Recall that the Baire kernel of A ⊂ R is a set B ⊂ A having the Baire property such that for each set C having the Baire property if C ⊂ A, then C\B ∈ I. For each set A ⊂ R there exists the Baire kernel of A (compare [11] or [6], §11.IV, Cor. 1, p. 90).

Definition 13
We shall say that a function f : R → R is intensely continuous at a point x 0 ∈ R if and only if x 0 is an intensity point of a Baire kernel of for each ε > 0. The right-or left intense continuity of f at x 0 is defined in an obvious way.

Definition 14
We shall say that a function f : The right-or left intense continuity as well as intense continuity I-almost everywhere is defined in an obvious way.

Theorem 15 A function f : R → R is intensely continuous (right intensely continuous) if and only if is intensely continuous (right intensely continuous) at each point x ∈ R.
Proof We shall prove first that if for each point x ∈ A there exists a set B x ⊂ A having the Baire property such that x ∈ i (B x ), then A has the Baire property. Suppose that it is not the case. Then there exists an interval (a, b) ⊂ R such that for each (c, d) ⊂ (a, b) the set A ∩ (c, d) is of the second category but not residual in (c, d). If B is an arbitrary Baire kernel of A ∩ (a, b), then B is of the first category, so for each x ∈ A ∩ (a, b) a point x is not an intensity point of B, a contradiction. The rest of the proof is routine. The proof in the case of right intense continuity is analogue.
To prove basic properties of intensely continuous functions we shall need some lemmas. In the sequel {(a i , b i )} i∈N will denote the basis of the natural topology on (0, 1).

Lemma 16
The sequence {n m } m∈N can be chosen to be increasing.
Proof This is obvious for m = 1. Suppose that the thesis holds for some m ∈ N. Let {r p } p∈N be an arbitrary increasing sequence of natural numbers such that r 1 > n m ≥ m. There exists a subsequence {r p k } k∈N such that lim k→∞ Since the intensity of a set at points different from zero is defined with the use of the translation, we obtain immediately:

Theorem 19 If a function f : R → R is right intensely continuous at each point, then f is of the first Baire class.
Proof Suppose that this is not the case. Then there exists a perfect set P ⊂ R and two real numbers a, b (a < b) such that the sets T 1 = {x ∈ P : f (x) < a} and T 2 = {x ∈ P : f (x) > b} are both dense in P in the natural topology (see for example [10] or [6], p. 395). Let C i < T i be a countable set, dense in T i (thus also dense in P) for i = 1, 2. Consider first the set From the assumption it follows that each point of T 1 is a right intensity point of T 1 . We shall show that if x ∈ O a ∩ P and G a is a regular open part of T 1 , then x is not a rarefaction point of G a . Indeed, for each m ∈ N there exists k m ∈ N such that x ∈ ∞ m=1 O m,k m . Let n m = n m (x k m ) be a number associated with m ∈ N and x k m ∈ C 1 . We can (and, choosing a subsequence if necessary, shall) suppose that {n m } m∈N is an increasing sequence. For each m ∈ N we have A point x 0 is a right-hand intensity point of T and is a right-hand rarefaction point neither of T 1 nor of T 2 . Hence T ∩ T 1 = ∅ and T ∩ T 2 = ∅. Let

Theorem 20 If f : R → R is intensely continuous at each point, then f is of the first Baire class and has the Darboux property.
Proof The first part follows from the previous theorem. The second is an immediate consequence of the following result of Z. Zahorski: if f : R → R is Baire one and for each a ∈ R the sets {x : f (x) > a} and {x : f (x) < a} are bilaterally dense in itself, then f has the Darboux property ( [16], see also [1], Th. 1.1 (8)).

Theorem 21 A function f : R → R has the Baire property if and only if it is intensely continuous I-almost everywhere.
Proof Suppose that f has the Baire property. Then there exists a residual set E ⊂ R such that the restriction f | E is continuous. Then f is intensely continuous at each point of E, so it is intensely continuous I-almost everywhere.
Suppose now that f is intensely continuous I-almost everywhere. Consider the set A = {x ∈ R : f (x) < a} for a ∈ R. Let E be the set of points where f is intensely continuos. Since A\E is of the first category, it is enough to show that E ∩ A has the Baire property. If x ∈ E ∩ A, then there exists a T i -neighbourhood , so E ∩ A ∈ T i and obviously it has the Baire property.

Restrictional intense continuity
Until now we have considered "topological" intense continuity of a function f : R → R at a point x 0 ∈ R. One can also consider a kind of "restrictional" or "path" intense continuity defined in the following way (compare [9] or [12]).

Definition 22
We say that a function f : R → R is restrictively intensely continuous at a point x 0 if and only if there exists a According to [5] if a topology T on the real line is invariant with respect to translations, then the topological and restrictional continuities (both with respect to T ) are equivalent if and only if the following condition (called (J 2 ) in [12]), pp. 28-30) holds: (J 2 ) For each descending sequence {U n } n∈N of right-hand T -neighbourhoods of 0 there exists a decreasing sequence {h n } n∈N tending to 0 such that {0}∪ ∞ n=1 (U n ∩ [h n+1 , h n )) is also a right-hand T -neighbourhood of 0 and the analogous condition for the left-hand case (with necessary changes) holds.
For the topology T i the above condition for the right-hand case can be formulated in the following more suitable way: For each ascending sequence {A n } n∈N of sets having the Baire property such that 0 is a right-hand rarefaction point of each A n there exists a decreasing sequence {h n } n∈N of positive numbers tending to 0 such that 0 is a right-hand rarefaction point of ∞ n=1 (A n ∩ [h n+1 , h n )). The formulation for the left-hand case is analogous. Observe that restrictional continuity for the above mentioned topologies always implies the topological continuity.
It is well known ([1], Th. 5.6, p. 23 or [12], Ex. 14.1, p. 27) that a function f : R → R is approximately continuous at a point x 0 if and only if it is restrictively approximately continuous at that point. In [9] it was proved that for I-approximately continuous the situation is different, namely the condition (J 2 ) does not hold. Now we shall prove that for intensely continuous functions restrictional and topological continuities at a point also are different because (J 2 ) fails for T i .

Proof For
, then x ∈ F n = E n,m p for each n such that k m p−1 ≤ n < k m p . Hence 1 So, if x ∈ lim sup p→∞ D p , then lim sup n→∞  Then E n,m = n(n + 1)(A m − 1 n+1 ) ∩ (0, 1) for n, m ∈ N and 0 is a right-hand rarefaction point of each A m , so the result follows immediately from the above lemma.

Corollary 25 The topology T i does not fulfil the condition (J 2 ).
Proof Let {A m } m∈N be a sequence of sets from Theorem 11 and let {h m } m∈N∪{0} be an arbitrary decreasing sequence convergent to 0. If {k m } m∈N∪{0} is an increasing sequence of positive integers such that 1 and the conclusion follows immediately from the above theorem.

More properties of the intense topology
It is well known that if A ⊂ [0, 1] is a regular open set such that 0 is a right-hand I-density point of A, then there exists a right interval set E ⊂ A such that 0 is also a right-hand I-density point of E (see, for example [3], Lemma 2.2.4, p. 25). We shall show that for intensity the situation is different.
Recall that a right interval set (open or closed) at a ∈ R is a set of the form ∞ n=1 (a n , b n ) (or ∞ n=1 [a n , b n ]) such that b n+1 < a n < b n for n ∈ N and a = lim n→∞ a n . A left interval set at a is defined in the same way. The set E is an interval set if it is the union of a right interval set and a left interval set at the same point.
Proof Each positive integer n can be represented uniquely in a form n = 2 k +l, where k ∈ N ∪ {0} and l ∈ {0, 1, . . . , 2 k − 1}. Let q ∈ (0, 1). Put E n = (0, 1)\ ∞ m=0 [q 2 k ·m+l+1 , q 2 k ·m+l ] for n = 2 k + l and A n = k i=0 (E n + i k+1 ) ∩ ( i k+1 , i+1 k+1 ) for n ∈ 2 k + l. Observe that for each k ∈ N and l ∈ {0, 1, . . . , 2 k−1 } we have 2 k +l n=2 k χ A n (x) ≥ l for all x ∈ (0, 1) except on the set i k+1 + q m : i ∈ {0, 1, . . . , k}, m ∈ N . Hence it is not difficult to see that lim n→∞ 1 n n j=1 χ A j (x) = 1 except on a set Q ∩ (0, 1), which means that the convergence is I-a.e. which is stronger than the convergence in category. We shall show that the subsequence n k = 2 k is a required subsequence. Indeed, let B n ⊂ A n for each n ∈ N be a set consisting of a finite number of components of A n . Observe that for each k ∈ N there exists a positive number ε k such that From this it follows immediately that if D = lim sup k→∞ D k , then lim inf k→∞ Obviously D is dense G δ set in (0, 1), so it is residual in (0, 1). The same holds for each subsequence of {n k } k∈N , which ends the proof.  , then h([a, b]) is also a closed interval and since h is one-to-one, it must be a homeomorphism h : (R, T nat ) → (R, T nat ). Suppose that h is increasing. In the remaining case the proof is similar.
Let A ⊂ R be a T i -open set such that 0 is a right-hand intensity point of A, but it is not a right-hand intensity point of any interval set included in the regular part G of A. Lemma 29 There exists a sequence {E n } n∈N of subsets of (0, 1 2 ) having the Baire property such that lim n→∞ 2n E n for n ∈ N, does not converge to 0 in category.
Observe now that for each x ∈ (0, 1) we have lim n→∞ 2n · E n = D k for 2 k−1 ≤ n < 2 k − 1. The set D = lim sup k→∞ D k is a G δ set dense in (0, 1 2 ). If x ∈ D, then for infinitely many k's x ∈ 2 k −1 n=2 k−1 F n = D k and then lim sup k→∞ For each subsequence {2 k p −1 } p∈N of the sequence {2 k − 1} k∈N we have also lim sup p→∞ 2 for x ∈D = lim sup p→∞ D k p and the setD is also a G δ set dense in (0, 1 2 ). Theorem 30 There exists a set A ⊂ (0, 1) such that 0 is a right-hand rarefaction point of A but not a right-hand rarefaction point of 1 2 A.
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