Chapter 23
Density type topologies generated by functions.
f
-density as a generalization of hsi-density and
ψ -density
TOMASZ FILIPCZAK
2010 Mathematics Subject Classification:54A10, 28A05, 26A15, 54C30 .
Key words and phrases:lower density operator, density topology, f -density, comparison of topologies.
The notions of a density point and an approximately continuous function have been defined at the beginning of XX century. The density topology was defined by Haupt and Pauc in the fifties ([17]) and re-invented by Goffman and Waterman in 1961 ([16]). Let us recall the basic notions. We write A ∼ B instead of λ (A 4 B) = 0. A point x ∈ R is called a density point of a set A ∈ L if
lim
h→0+
λ (A ∩ [x − h, x + h])
2h = 1.
The set of all density points of a set A ∈ L we denote by Φd(A). The operator
Φdis a lower density operator and a family
Td:= {A ∈ L : A ⊂ Φd(A)}
is a topology called the density topology.
Over the last thirty years several density-type topologies has been studied by many mathematicians. All such topologies are generated by operators called lower density operators or slight different operators (studied by Hejduk in [19] and called almost density operators).
We start by describing the properties of density-type topologies that arise from the definition of a lower density operator and an almost lower density operator in the sense of Hejduk. We recall definitions of an hsi-density topol-ogy and a ψ-density topoltopol-ogy (described in the previous chapter). Then we define an f -density topology which is a generalization of several density-type topologies. In particular, hsi-density topologies and ψ-density topologies are f-density topologies. At first sight these topologies have quite different proper-ties. However, they can be studied as special cases of the more general concept. For any A ⊂ R we denote by KA a measurable kernel of A, and we set
−A := {−a : a ∈ A} and A + x := {a + x : a ∈ A}, x ∈ R.
23.1 Density-type topologies consisting of measurable sets
We will study operators Φ : L → P (R). Consider the following properties: (D0) if A ∈ Tnatthen A ⊂ Φ (A),
(D1) Φ ( /0) = /0 and Φ (R) = R, (D2) Φ (A ∩ B) = Φ (A) ∩ Φ (B), (D3) if A ∼ B then Φ (A) = Φ (B), (D4) Φ (A) ∼ A and (D4’) λ (Φ (A) \ A) = 0 for measurable A and B.
If Φ fulfils (D1)-(D4) then it is called a lower density operator. An operator fulfilling (D1)-(D3) and (D4’) has been named by Hejduk an almost lower density operator.
Remark 23.1. If Φ fulfils (D2) then Φ is monotonic i.e. (D2’) if A ⊂ B then Φ (A) ⊂ Φ (B).
Remark 23.2. From (D2’) and (D3) it follows that for any A, B ∈ L such that λ (A \ B) = 0 we have Φ (A) ⊂ Φ (B).
We define
TΦ:= {A ∈ L : A ⊂ Φ (A)}.
Remark 23.3. If Φ1(A) ⊂ Φ2(A) for A ∈ L then TΦ1 ⊂ TΦ2. Consequently, if
We will prove that if an operator Φ satisfies conditions (D0)-(D3) and (D4’) then TΦ forms a topology. We will examine properties of such operators and
topologies generated by them. We will also consider operators satisfying the condition (D4), stronger than (D4’). To make the text self-contained we shortly repeat some proofs known in the literature. A lot of proofs are based on con-siderations for density topology (compare [23] and [29]). Some properties can be proved under the weaker assumptions (see [22] and [19]). Recall that N stands for the σ -ideal of null sets on the real line.
Theorem 23.4. If Φ : L → P (R) fulfils (D0)-(D3) and (D4’) then (1)TΦ forms a topology on R,
(2)intTΦE⊂ E ∩ Φ (KE) for E ⊂ R,
(3) if A∈ N then A is TΦ-closed,TΦ-nowhere dense andTΦ-discrete,
(4) A∈ N if and only if A is TΦ-closed andTΦ-discrete, (5) A is aTΦ-compact set if and only if A is finite,
(6) the space(R, TΦ) is neither first countable, nor Lindelöf, nor separable,
(7)cardTΦ= 2c,
(8)Tnat $ TΦ,
(9) A∈ L if and only if A is a TΦ-Borel set, (10)(R, TΦ) is a Hausdorff space,
(11)(R, TΦ) is not a normal space.
Proof. (1) Of course /0 ∈ TΦ. If A, B ∈ TΦ then by (D2), A ∩ B ⊂ Φ (A) ∩
Φ (B) = Φ (A ∩ B), which implies A ∩ B ∈ TΦ. Assume that At ∈ TΦ for
t∈ T and A :=S
t∈TAt. Since At\ KA∈ N , Remark 23.2 shows that Φ (At) ⊂
Φ (KA), and consequently KA⊂ A = [ t∈T At ⊂ [ t∈T Φ (At) ⊂ Φ (KA) .
From (D4’) it follows that A ∼ KA, so A ∈ L. Using (D2’) again, we conclude
that A ⊂ Φ (KA) ⊂ Φ (A), which gives A ∈ TΦ.
(2) Since intTΦE is a measurable subset of E, intTΦE\ KE∈ N . From Remark
23.2 we obtain intTΦE⊂ Φ (intTΦE) ⊂ Φ (KE).
(3) Let A ∈ N . By (D3) and (D1), R \ A ⊂ R = Φ (R) = Φ (R \ A), hence A is TΦ-closed. The set A is nowhere dense because from (2), (D1) and (D3) we
have intTΦA⊂ Φ (A) = /0. If x ∈ A then (R \ A) ∪ {x} ∈ TΦ, and consequently
{x} is TΦ-open in A. Thus A is TΦ-discrete.
(4) Suppose that A is TΦ-closed and TΦ-discrete. For any x ∈ A there exists a
x∈ Ux⊂ Φ (Ux) = Φ (Ux\ {x}) ⊂ Φ (R \ A) ,
and so A ⊂ Φ (R \ A). Therefore A ⊂ Φ (R \ A) \ (R \ A) ∈ N , by (D4’). (5) Suppose that A is infinite. Let B ⊂ A be a countable infinite subset of A. Then {(R \ B) ∪ {x}}x∈Bis a TΦ-open cover of A without finite subcover.
(6) If An, n ∈ N, are TΦ-open neighbourhoods of x and xn∈ An\ {x} then the
set A1\ {xn: n ∈ N} is a TΦ-open neighbourhood of x which does not include
any An. Thus (R, TΦ) is not first countable.
Let C be the Cantor ternary set. Then {(R \C) ∪ {x}}x∈Cis a TΦ-open cover of
Awithout countable subcover, and consequently (R, TΦ) is not Lindelöf. Since
any countable set is TΦ-closed, (R, TΦ) is not separable.
(7) Any subset of the Cantor set C is TΦ-closed. Hence cardTΦ≥ cardP (C) =
2c.
(8) By (D0), Tnat ⊂ TΦ. Moreover (R \ Q) ∈ TΦ\ Tnat.
(9) The inclusion BorTΦ ⊂ L follows from TΦ ⊂ L. From (3) and (8) we
con-clude that N ⊂ BorTΦand BorTnat⊂ BorTΦ. Thus we have L ⊂ BorTΦ, because
every measurable set is a sum of a Borel set and a null set. (10) It follows from (8).
(11) (compare [22, Prop. 7.17]) Suppose, contrary to our claim, that (R, TΦ)
is a normal space. Write F := {C ∩ Φ (A) : A ∈ L}, where C is the Cantor set. For any measurable set A there is a Borel set A0 such that A ∼ A0, so card {Φ (A) : A ∈ L} ≤ c. Therefore, one can find a set H ⊂ C such that H /∈ F . Since H and F := C \ H are TΦ-closed, there are disjoint TΦ-open sets AH, AF
containing H and F. Thus Φ (AH) ∩ Φ (AF) = Φ (AH∩ AF) = /0 and
H⊂ C ∩ AH⊂ C ∩ Φ (AH) ⊂ C \ Φ (AF) ⊂ C \ AF⊂ C \ F = H.
Hence H = C ∩ Φ (AH), which gives a contradiction, because C ∩ Φ (AH) ∈ F .
u t Remark 23.5. The assumption (D0) has not been used in the proof of condi-tions (1)-(7) and (11).
If Φ fulfils additionally the condition (D4), we obtain stronger results. Theorem 23.6. If Φ : L → P (R) fulfils (D0)-(D4) then
(1) Φ (A) ∈ L for A ∈ L,
(2) Φ (Φ (A)) = Φ (A) for A ∈ L, (3)TΦ = {Φ (A) \ N : A ∈ L, N ∈ N }, (4)intTΦE= E ∩ Φ (KE) for E ⊂ R,
(6) Φ (A) = intTΦ(clTΦA) for A ∈ L,
(7) A∈ N ⇔ A is TΦ-nowhere dense⇔ A is TΦ-meager,
(8) A∈ L ⇔ A has TΦ-Baire property⇔ A is a sum of a TΦ-open set and a
TΦ-closed set,
(9)(R, TΦ) is a Baire space.
Proof. (1)-(2) From (D4) we have Φ (A) ∼ A. Hence Φ (A) ∈ L and Φ (Φ (A)) = Φ (A), by (D3).
(3) If A ∈ TΦ then A = Φ (A) \ (Φ (A) \ A), and by (D4), Φ (A) \ A ∈ N .
Let A ∈ L and N ∈ N . From (D3) and (2) it follows that Φ (Φ (A) \ N) = Φ (Φ (A)) = Φ (A) ⊃ Φ (A) \ N, which gives Φ (A) \ N ∈ TΦ.
(4) The inclusion intTΦE ⊂ E ∩ Φ (KE) follows from Theorem 23.4. Let
x∈ E ∩ Φ (KE) and A := KE∪ {x}. The set A ∩ Φ (A) is a TΦ-open
neigh-bourhood of x, because
Φ (A ∩ Φ (A)) = Φ (A) ∩ Φ (Φ (A)) = Φ (A) ⊃ A ∩ Φ (A) . Since A ∩ Φ (A) ⊂ A ⊂ E, we have x ∈ intTΦE.
(5) From (D4) and (4) we conclude that intTΦA= A ∩ Φ (A) ∼ A and clTΦA=
R \ intTΦ(R \ A) ∼ R \ (R \ A) = A.
(6) For any measurable set A we have A ∼ clTΦAand clTΦA= R\intTΦ(R \ A) ⊃
R \ Φ (R \ A) ⊃ Φ (A). Hence
intTΦ(clTΦA) = clTΦA∩ Φ (clTΦA) = clTΦA∩ Φ (A) = Φ (A) .
(7) According to Theorem 23.4, it is sufficient to prove that each TΦ-nowhere
dense set is a null set. Let A be a TΦ-nowhere dense set. Then clTΦAis also
TΦ-nowhere dense. Using (6) we get Φ (A) = intTΦ(clTΦA) = /0, which yields
A∼ /0.
(8) If A ∈ L then A = (A ∩ Φ (A)) ∪ (A \ Φ (A)), A ∩ Φ (A) is TΦ-open and
A\ Φ (A) is TΦ-closed (because it is a null set). Of course, a sum of a TΦ-open
set and a TΦ-closed set has TΦ-Baire property. Finally, if A = U 4 N where U
is TΦ-open and N is TΦ-meager, then U ∈ L and N ∈ N , by (7). Consequently,
Ais measurable.
(9) If A is TΦ-open and TΦ-meager, then A ⊂ Φ (A) and A ∈ N , so A = Φ (A) =
/0. ut
Remark 23.7. From (6) it follows that A is TΦ-regular open if and only if
Φ (A) = A.
In the following chapter there are constructed an operator Φ, satisfying (D0)-(D3) and (D4’), and closed sets of positive measure F0, F1 such that
Φ (F0) = /0 and Φ (F1) is a singleton. In fact, it is proved that for any
func-tion f with lim infx→0+ f(x)x = 0 there are closed sets of positive measure such
that Φf(F0) = /0 and Φf(F1) = {0}, where Φf is an f -density operator defined
in the section 5 (see Theorem 24.6 and Theorem 24.7 in chapter 24). Using this result one can easy check that properties (2)-(9) from Theorem 23.6 can be false if we replace (D4) by (D4’).
Note that in the following theorem items start from (2) to stress the similar-ity to properties of Theorem 23.6.
Theorem 23.8. If Φ : L → P (R) fulfils (D0)-(D3) and (D4’) and F0, F1 are
closed sets of positive measure such that Φ(F0) = /0, Φ(F1) = {0} then
(2) Φ (Φ (F1)) = /0 6= Φ (F1),
(3) Φ (F1) /∈ TΦ,
(4)intTΦF1= /0 6= F1∩ Φ (F1),
(5) λ (F0\ intTΦF0) = λ (F0) > 0 and λ (clTΦ(R \ F0) \ (R \ F0)) = λ (F0) > 0,
(6) Φ (F1) = {0} 6= intTΦclTΦF1,
(7) F0 isTΦ-nowhere dense, but is not a null set; R is TΦ-meager, but is not
TΦ-nowhere dense,
(8) each nonmeasurable set is TΦ-meager, and consequently has TΦ-Baire
property,
(9)(R, TΦ) is not a Baire space.
Proof. Conditions (2)-(6) are clear. Obviously, F0 is TΦ-nowhere dense. By
Smital’s Lemma, the set E :=S
q∈Q(F0+ q) has a full measure (compare [20],
p. 65). Thus R \ E is TΦ-nowhere dense. Since every F0+ q is TΦ-nowhere
dense too, R is TΦ-meager. This implies (8)-(9). ut
Now we construct an easy example of an operator Φ : L → P (R), satisfy-ing (D0)-(D3) and (D4’), such that the set Φ (A) need not be measurable, for measurable A.
Example 23.9.Suppose that Φ : L → P (R) fulfils (D0)-(D3) and (D4’), F0is a closed set of positive measure such that Φ(F0) = /0 and D is a nonmeasurable
subset of F0. Write
b
Φ (A) := (Φ (A) \ D) ∪ (Φd(A) ∩ D)
for A ∈ L. It is easy to check that bΦ fulfils (D0)-(D3) and (D4’). But bΦ (F0) =
23.2 hsi-density
Reminding the concept of ordinary density point it is worth observing that x∈ Φ (A) if and only if
lim n→∞ λ A ∩x −1n, x +1n 2 n = 1.
Replacing the sequence 1nn∈Nby a fixed sequence (sn)n∈Ndecreasingly
tend-ing to zero, we obtain the notion of a density generated by the sequence (sn)n∈N.
Denote by eS the family of all nonincreasing and tending to zero sequences of positive numbers and fix hsi = (sn)n∈N∈ eS and A ∈ L. If
lim
n→∞
λ (A ∩ [x − sn, x + sn])
2sn
= 1 then x is called an hsi-density point of a set A. Analogously, if
lim n→∞ λ (A ∩ [x, x + sn]) sn = 1 lim n→∞ λ (A ∩ [x − sn, x]) sn = 1
then we say that x is a right-hand (left-hand) hsi-density point of A. The set of all hsi-density points (right-hand, left-hand hsi-density points) of a set A we denote by Φhsi(A) (Φhsi+ (A), Φhsi− (A), respectively). Clearly, Φhsi(A) =
Φhsi+ (A) ∩ Φhsi− (A). We will write Thsiinstead of TΦhsi, i.e.
Thsi=A ∈ L : A ⊂ Φhsi(A) .
Obviously, Td= Th1
ni and Td
⊂ Thsifor any hsi ∈ eS.
The notion of an hsi-density point has been defined in [12]. Properties of an operator Φhsiand a topology Thsihave been studied also in [11], [13] and [21].
Note that the authors used nondecreasing sequences tending to ∞ instead of nonincreasing sequences tending to zero, and considered s1
n instead of sn. Let
us recall the basic properties of Φhsiand Thsi.
Theorem 23.10 ([12]). For any sequence hsi ∈ eS (1) Φhsisatisfies (D0)-(D4),
(2)Thsiis a topology,
Theorem 23.11 ([12]). Let hsi ∈ eS. (1) Iflim infn→∞sn+1sn > 0 then Thsi= Td.
(2) Iflim infn→∞sn+1sn = 0 then Thsi% Td.
The density topology is invariant under translation and under multiplication by nonzero numbers. However, hsi-density topologies bigger than Td are not
invariant under multiplication.
Theorem 23.12 ([12]). Let hsi ∈ eS and m ∈ R. (1) The topologyThsiis invariant under translation.
(2) If |m|≥ 1 then Thsiis invariant under multiplication by m.
(3) IfThsi% Td and |m|< 1 then Thsiis not invariant under multiplication by
m.
All hsi-density topologies fulfil the same separating axioms as Td.
Theorem 23.13 ([21]). For any hsi ∈ eS the space R, Thsi is completely
regu-lar but not normal.
Theorem 23.14 ([13]). Let hsi ∈ eS. A set A is connected in R, Thsi if and only
if A is connected in(R, Tnat).
Despite the fact that hsi-density topologies have very similar properties, there are a lot of nonhomeomorphic hsi-density topologies.
Theorem 23.15 ([13]). Suppose that Thsi6= Td6= Thti.
(1) The spaces(R, Td) and R, Thsi are not homeomorphic.
(2) IfThsi* ThtiandThti* Thsithen the spaces R, Thsi and R,Thti are not homeomorphic.
(3) For any m> 1, Th1
msi $ Thsi$ Thmsi
and the spaces
R, Th1 msi
, R, Thsi,
R, Thmsi are homeomorphic.
23.3 ψ-density
In [25] Terepeta and Wagner-Bojakowska, based on the results of Taylor ([24]), introduced the notion of a ψ-density point. They defined a ψ-density operator and a ψ-density topology, and studied the basic properties of these notions. The other interesting results can be found in [27], [28], [1], [26] and [14]. Some of them are presented in the previous chapter of this book.
Recall that bC denotes the family of all continuous and nondecreasing func-tions ψ : (0, ∞) → (0, ∞) with limx→0+ψ (x) = 0. We say that x is a ψ -density pointof a measurable set A if
lim
h→0+
λ ([x − h, x + h] \ A)
2hψ (2h) = 0.
The set of all ψ-density points of A is denoted by Φψ(A). We write Tψ instead
of TΦψ i.e.
Tψ =A ∈ L : A ⊂ Φψ(A) .
The Second Taylor’s Theorem (see section 22.2 in the previous chapter) im-plies:
Theorem 23.16. For each ψ ∈ bC there exists a set E such that λ E \ Φψ(E)
is positive.
Therefore, the Lebesgue Density Theorem does not hold for ψ-density, and no Φψ is a lower density operator. However, Φψ(A) ⊂ Φd(A), so
λ Φψ(A) \ A = 0
for A ∈ L. We also have:
Proposition 23.17 ([25]). For any ψ ∈ bC and A ∈ L, Φψ(A) is the set of type
Fσ δ.
Theorem 23.18 ([25]). For any ψ ∈ bC (1) Φψ satisfies (D0)-(D3) and (D4’),
(2)Tψ is a topology,
(3) Φψ andTψ satisfy all conditions of Theorem 23.4.
Moreover, for any ψ ∈ bC, Tnat $ Tψ $ Td and the family of Tψ-connected
sets is the same as in (R, Tnat) (compare [25]). Since Φψ does not satisfy
(D4), there are several differences between the density topology and ψ-density topologies. For example, the space R, Tψ is not regular ([2]) and it is not a
Baire space ([26]). More properties of ψ-density topologies are described in section 22.3 in the previous chapter.
23.4 The definition and basic properties of f -density
(A1) limx→0+f(x) = 0,
(A2) lim infx→0+ f(x)x < ∞,
(A3) f is nondecreasing.
Let f ∈ A, A ∈ L and x ∈ R. We say that x is a righthand (lefthand) f -density pointof A if lim h→0+ λ ([x, x + h] \ A) f(h) = 0 lim h→0+ λ ([x − h, x] \ A) f(h) = 0 .
By Φ+f (A) (Φ−f (A)) we denote the set of all right-hand (left-hand) f -density points of A. If x ∈ Φf(A) := Φ+f (A) ∩ Φ
−
f (A) then we say that x is an f -density
pointof A.
Remark 23.19. Condition (A2) is essential because limh→0+ f(h)h = ∞ implies λ ([x,x+h]\A)
f(h) ≤
h
f(h) h→0+→ 0.
The definition of f -density was introduced in [3]. In this paper continuity of functions from the family A was assumed. In subsequent papers this condition was omitted (compare Theorem 23.34). Considering functions from the family A, we often define f (x) only for x ∈ (0, δ ), for some δ > 0. To distinguish the notion of f -density from ψ-density (considered in the previous section) we will use Latin letters ( f , g) defining density generated by function and Greek letters (ψ) in the second case.
Straightforward from the properties of Lebesgue measure it follows: Proposition 23.20. Φf(A + x) = Φf(A) + x for f ∈ A, A ∈ L and x ∈ R.
Since functions from the family A are monotonic, we can describe an f -density point in equivalent way.
Proposition 23.21 ([3]). Let f ∈ A, A ∈ L and x ∈ R. Then x is an f -density point of A if and only if
lim
h→0,k→0 h≥0,k≥0,h+k>0
λ ([x − h, x + k] \ A) f(h + k) = 0. Proof. Sufficiency is evident. Since
λ ([x − h, x + k] \ A) f(h + k) ≤ λ ([x − h, x] \ A) f(h) + λ ([x, x + k] \ A) f(k)
Theorem 23.22 ([3]). If f ∈ A and A ∈ L then Φf(A), Φ+f (A) and Φ−f (A) are
the sets of type Fσ δ.
Proof. For any h > 0 the function Fh(x) = λ ([x−h,x]\A)f(h) is continuous. Thus
the-orem follows from the equality Φ+f (A) = \ n∈N [ δ ∈Q+ \ h∈(0,δ ) Fh−1 0,1 n . u t The following theorem states that an f -density point of a set A can not be a dispersion point of A. Thus an operator Φf satisfies the condition (D4’).
Theorem 23.23 ([3]). Let f ∈ A, A ∈ L and x ∈ R. (1) If x∈ Φ+f (A) then lim infh→0+λ ([x,x+h]\A)h = 0.
(2) Φf(A) ⊂ R \ Φd(R \ A).
Proof. Suppose that lim infh→0+λ ([x,x+h]\A)h > 0. From (A2) it follows that
lim sup h→0+ λ ([x, x + h] \ A) f(h) ≥ lim infh→0+ λ ([x, x + h] \ A) h · lim suph→0+ h f(h) > 0 which gives x /∈ Φ+f (A). The second condition is a consequence of the first.
u t Theorem 23.24 ([3]). Let f ∈ A. The operator Φf fulfils (D0)-(D3) and (D4’)
i.e. for any A, B ∈ L we have: (0) if A∈ Tnatthen A⊂ Φf(A),
(1) Φf( /0) = /0 and Φf(R) = R,
(2) Φf(A ∩ B) = Φf(A) ∩ Φf(B),
(3) if A∼ B then Φf(A) = Φf(B),
(4’) λ (Φf(A) \ A) = 0.
Proof. Conditions (0), (3), Φf(R) = R and Φf(A ∩ B) ⊂ Φf(A) ∩ Φf(B) are
obvious. The equality Φf( /0) = /0 follows from (A2). Since
λ (I \ (A ∩ B)) ≤ λ (I \ A) + λ (I \ B)
for every interval I, we have Φf(A) ∩ Φf(B) ⊂ Φf(A ∩ B). From Theorem
23.23 we conclude that
Thus the Lebesgue Density Theorem implies (4’). ut The family TΦf will be denoted by Tf, i.e.
Tf =A ∈ L : A ⊂ Φf(A) .
According to Theorem 23.24, Φf and Tf fulfil all conditions of Theorem 23.4:
Theorem 23.25. For any f ∈ A (1)Tf forms a topology on R,
(2)intTfE⊂ E ∩ Φf(KE) for E ⊂ R,
(3) if A∈ N then A is Tf-closed,Tf-nowhere dense andTf-discrete,
(4) A∈ N if and only if A is Tf-closed andTf-discrete,
(5) A is aTf-compact set if and only if A is finite,
(6) the space(R, Tf) is neither first countable, nor Lindelöf, nor separable,
(7)cardTf = 2c,
(8)Tnat $ Tf,
(9) A∈ L if and only if A is a Tf-Borel set,
(10)(R, Tf) is a Hausdorff space,
(11)(R, Tf) is not a normal space.
In general, there are two possibilities to define density points. We can do it "in a symmetric way", examining the set on intervals centered at x, or "in one-sided way" - on intervals [x − h, x] and [x, x + h]. In this sense, the definition of f -density is "one-sided". However, "symmetric" definition leads us to the same notion (using, if necessary, another function).
Let f ∈ A, A ∈ L and x ∈ R. We say that x is a symmetric f -density point of Aif lim h→0+ λ ([x − h, x + h] \ A) f(2h) = 0. By Φs
f(A) we denote the set of all symmetric f -density points of A. For any
function f defined on (0, ∞) we set f∗(x) := f (2x). An easy verification shows that: Proposition 23.26 ([15]). (1) f ∈ A if and only if f∗∈ A, (2) Φf∗ = Φsf for f ∈ A, (3)Φf : f ∈ A = n Φsf : f ∈ A o .
Example 23.27.Let f (x) :=n!1 for x ∈ 1 (n+1)!, 1 n! i and A:= (−∞, 0] ∪ ∞ [ n=2 1 (n + 1)!, 1 2 · n! .
Clearly, f ∈ A. It is easy to check that λ ([−h,h]\A)f(2h) ≤ 1
nfor h ∈ 1 (n+1)!, 1 n! i , and consequently, 0 ∈ Φsf(A). Butλ([0,
1 n!]\A)
f(1 n!)
>12, which gives 0 /∈ Φf(A).
It should be mentioned that the notion of f -density could be defined even more generally. In [18] Hejduk considered f -density points (and symmetric
f-density points) assuming only condition (A2).
Theorem 23.28 ([18]). Let f : (0, ∞) → (0, ∞) be a function satisfying lim inf
x→0+
f(x) x < ∞. Then
(1) Φf and Φsf satisfy conditions (D0)-(D3) and (D4’),
(2)Tf andTfsform topologies such thatTnat⊂ Tfs⊂ Tf.
23.5 The comparison of f -density topologies
Let f , g ∈ A. We ask for conditions under which the inclusion Tf ⊂ Tgholds.
The necessary and sufficient condition is presented in the following chapter (Theorem 24.2). Now, we show an easy sufficient condition. We also formu-late a necessary and sufficient condition to compare Tf with Td. Using it, we
divide the family of all f -density topologies into two parts. The first consists of topologies similar to hsi-density topologies and the second - similar to ψ-density topologies.
Obviously, different functions can generate the same operator (for exam-ple Φf = Φ2 f). Fortunately, different operators generate different topologies.
Moreover, to prove the inclusion Tf ⊂ Tg, it is enough to show that the
condi-tion 0 ∈ Φ+f (A) implies 0 ∈ Φg+(A).
Theorem 23.29 ([5], [9]). For each f , g ∈ A the following conditions are equivalent (1)∀A∈L 0 ∈ Φ+f (A) ⇒ 0 ∈ Φ+ g (A) ,
(2)∀A∈L (0 ∈ Φf(A) ⇒ 0 ∈ Φg(A)),
(3)∀A∈LΦf(A) ⊂ Φg(A),
(4)Tf ⊂ Tg.
Proof. Implication (1)⇒(2) follows from 0 ∈ Φ−f (A) ⇔ 0 ∈ Φ+f (−A). By x ∈ Φf(A) ⇔ 0 ∈ Φf(A − x), we obtain (2)⇒(3). Implication (3)⇒(1) is clear.
Assume now that Tf ⊂ Tg, 0 ∈ Φ+f (A) but 0 /∈ Φg+(A). We can find ε > 0 and
a decreasing sequence (xn) tending to zero such that
λ ([0, xn] \ A)
g(xn)
≥ ε for every n. Defining an= xn− λ ((xn+1, xn) \ A) and
B:= (−∞, 0] ∪ ∞ [ n=1 (xn+1, an) we obtain that 0 /∈ Φ+
g (B). On the other hand 0 ∈ Φ+f (B), because
λ ([0, x] \ B) ≤ λ ([0, x] \ A)
for any x > 0. Consequently, B ∈ Tf \ Tg, which gives a contradiction. ut
According to the latter theorem, we will usually formulate conditions for topologies and operators, and will prove only the condition 23.29 from Theo-rem 23.29.
Let f , g ∈ A. We say that f precedes g if lim supx→0+g(x)f(x)< ∞. We denote it by f ≺ g. If lim supx→0+g(x)f(x)= ∞ we write f ⊀ g (compare [25]).
Theorem 23.30 ([3]). Let f , g ∈ A. If f ≺ g then Tf ⊂ Tg.
Proof. Suppose that 0 ∈ Φ+f (A). Then
lim sup h→0+ λ ([0, h] \ A) g(h) ≤ lim suph→0+ λ ([0, h] \ A) f(h) ·lim suph→0+ f(h) g(h)= 0·lim suph→0+ f(h) g(h)= 0,
which means that 0 ∈ Φg+(A). ut
Example 23.31([5]). There exist f , g ∈ A such that f ⊀ g and Tf ⊂ Tg. Let
f(x) :=n!1 for x ∈ h 1 (n+1)!; 1 n! and g(x) :=n!1 for x ∈ 1 (n+1)!; 1 n! i . Then f ≥ g and f ⊀ g because f( 1 n!) g(1 n!)
x6= 1
n!, to prove 0 ∈ Φ +
g (A) it is enough to show that limn→∞
λ([0,n!1]\A) g(1 n!) = 0. For n ≥ 2 we have 0 ≤ λ 0, 1 n! \ A g n!1 ≤ λ 0,1+n!1 \ A + 1 (n!)2 f 1+n!1 = λ 0,1+n!1 \ A f 1+n!1 + 1 n! −→n→∞0,
which completes the proof.
Example 23.31 shows that Theorem 23.30 can not be reversed. Fortunately, if one of considered topologies is Td, the opposite implication also holds.
Theorem 23.32 ([5], [10]). For any f ∈ A we have (1)Tf ⊂ Tdif and only iflim supx→0+
f(x) x < ∞.
(2)Td⊂ Tf if and only iflim infx→0+ f(x)x > 0.
Proof. By Theorem 23.30, inequalities under consideration are sufficient con-ditions for inclusions of topologies. Suppose that lim supx→0+ f(x)x = ∞. Let A:=S∞
n=1(2xn+1, xn), where (xn) is a decreasing sequence of positive
num-bers such that 2xn+1 < xn < f(xn) n . Since λ (A∩[0,2xn]) 2xn < 1 2, 0 /∈ Φ + d (A). But
0 ∈ Φ+f (A), because for h ∈ (xn+1, xn] we haveλ ([0,h]\A)f(h) ≤ f2x(xn+1
n+1)<
2 n+1.
Sup-pose now that lim infx→0+ f(x)x = 0. Let A :=
S∞ n=1(xn+1, xn− f (xn)) where xn+1< f (xn) <1nxn. Then λ (A ∩ [0, xn]) xn >xn− f (xn) − xn+1 xn > 1 −2 n, and we easily obtain 0 ∈ Φd+(A). But 0 /∈ Φ+f (A) since λ ([0,xn]\A)
f(xn) ≥ 1. ut
Corollary 23.33. Let f ∈ A.
(1)Tf = Tdif and only if0 < lim infx→0+ f(x)x ≤ lim supx→0+ f(x)
x < ∞.
(2)Tf $ Tdif and only if0 = lim infx→0+ f(x)x ≤ lim supx→0+ f(x)
x < ∞.
(3)Td$ Tf if and only if0 < lim infx→0+ f(x)x < lim supx→0+ f(x)
x = ∞.
(4)Tf * TdandTd* Tf if and only if0 = lim inff(x)x < lim supx→0+ f(x)x = ∞.
Thus the family A splits into four subfamilies. However, it turns out that properties of the topology Tf depend mainly on whether id ≺ f . Therefore we
A1:= { f ∈ A : id ≺ f } = f ∈ A : lim inf x→0+ f(x) x > 0 , A0:= { f ∈ A : id ⊀ f } = f ∈ A : lim inf x→0+ f(x) x = 0 .
23.6 f -density and hsi-density
In this section we will describe a relation between hsi-density and f -density for f ∈ A1. First of all, we will show that any operator Φf generated by f ∈ A
is generated by some continuous function f1∈ A and by some function f2∈ A
constant on intervals. By Acwe denote the family of continuous functions from
A, and by Asthe family of functions from A for which there exist decreasing
and tending to 0 sequences (xn) and (an) such that f (x) = anfor x ∈ (xn+1, xn].
Theorem 23.34 ([5], [6]).
Φf : f ∈ A = Φf : f ∈ Ac = Φf : f ∈ As .
Proof. Let f ∈ A and a := f (1). As f is nondecreasing, the set f−1 2n+1a ,
a 2n
is either empty or is an interval (may be degenerated). Let (kn) be an increasing
sequence of all numbers n for which In:= f−1 2kn+1a ,
a
2kn is a
nondegener-ated interval, and let xn be the right endpoint of In. Evidently, xn+1 is the left
endpoint of In. Let us denote
g(x) := a
2kn for x ∈ (xn+1, xn] .
Obviously, g ∈ As. It is not difficult to prove that Φf = Φg(see [5], Th. 1).
Assume now that f ∈ As, i.e. f (x) = an for x ∈ (xn+1, xn], where (xn) and
(an) are decreasing sequences tending to 0 and such that lim inf n→∞ an xn < ∞. Put δn:= min xn−xn+1 2 , an+1 n and g(x) := anfor x ∈ [xn+1+ δn, xn] , linear on [xn+1, xn+1+ δn] .
Of course, g ∈ Ac. One can check that Φf = Φg(see [5, Th. 1]). ut
Now we are in the position to prove that density operators and hsi-density topologies are specific cases of f -hsi-density operators and f -hsi-density topologies. Let
fhsi(x) :=
snfor x ∈ (sn+1, sn] ,
for hsi ∈ eS. Obviously, fhsi∈ A and fhsi(x) ≥ x for every x.
Theorem 23.35 ([5]).
(1) If hsi∈ eS then fhsi∈ Asand Φhsi= Φfhsi.
(2) n Φhsi: hsi ∈ eS o ⊂ Φf : f ∈ A1 .
Proof. The condition fhsi∈ Asis obvious. To prove that Φhsi= Φfhsi it is
suf-ficient to show that for any measurable A 0 ∈ Φhsi+ (A) ⇔ 0 ∈ Φ+f
hsi(A) .
The first implication follows from the inequality λ ([0,x]\A)f(x) ≤ λ ([0,sn]\A)
sn for
x∈ (sn+1, sn], and the second from the equality f (sn) = sn. The inclusion (2)
follows immediately from the definition of fhsi. ut
Proposition 23.36 ([8]). For any function f ∈ A, there is a sequence hsi ∈ eS such thatTf ⊂ Thsi.
Proof. By (A2), there exist a sequence hsi ∈ eS and a positive number M such that f(sn)
sn < M for every n. It is easily seen that Tf ⊂ Thsi. ut
Using this inclusion we can characterize the family of Tf-connected sets.
Theorem 23.37 ([8]). For any f ∈ A, the family of Tf-connected sets is equal
to the family of sets connected in the natural topology.
Proof. The latter proposition shows that Tnat ⊂ Tf ⊂ Thsi for some hsi ∈ eS.
Thus our claim follows from Theorem 23.14. ut
Recall that any Φhsiis a lower density operator. The same is true for Φf with
f∈ A1.
Theorem 23.38 ([4]). If f ∈ A1then Φf satisfies (D4).
Proof. By Theorem 23.32, Td⊂ Tf. Hence Φd(A) ⊂ Φf(A) for A ∈ L, and
consequently, A \ Φf(A) ⊂ A \ Φd(A), which gives λ (A \ Φf(A)) = 0. This
completes the proof, because by Theorem 23.24, Φf fulfils (D4’). ut
By Theorem 23.22, Φf(A) ∈ L for each f ∈ A and A ∈ L. From Theorem 23.6
Theorem 23.39. If f ∈ A1then
(1) Φf(Φf(A)) = Φf(A) for A ∈ L,
(2)Tf =
Φf(A) \ N : A ∈ L, N ∈ N ,
(3)intTfE= E ∩ Φf(KE) for E ⊂ R,
(4)intTfA∼ A ∼ clTfA for A∈ L,
(5) Φf(A) = intTfclTfA for A∈ L,
(6) A∈ N ⇔ A is Tf-nowhere dense⇔ A is Tf-meager,
(7) A∈ L ⇔ A has Tf-Baire property ⇔ A is a sum of a Tf-open set and a
Tf-closed set,
(8)(R, Tf) is a Baire space.
Theorem 23.40 ([3]). If f ∈ A and α ≥ 1 then Tf is invariant under
multipli-cation by α. Proof. From λ ([0,h]\α A)f(h) =α λ([0, h α]\A) f(ααh) ≤ α λ([0,h α]\A)
f(αh) it follows that 0 ∈ Φf(A)
implies 0 ∈ Φf(αA). Let A ∈ Tf and x ∈ αA. Then
0 ∈ A − x α ⊂ Φf A− x α ⊂ Φf(αA − x)
and consequently, 0 ∈ intTf(αA − x) = intTf(αA) − x, which ends the proof.
u t Theorems formulated up to now show that topologies Tf generated by
f ∈ A1 have properties similar to the properties of topologies generated by
sequences. Any hsi-density topology is completely regular. In the next chapter it will be proved that f -density topologies are completely regular for f ∈ A1 (Theorem 24.15). There is a natural question if there exists an f -density topol-ogy, generated by a function from A1, which is not generated by a sequence. The positive answer was obtained in [5]. However, the example presented in this paper is quite complicated and the proof of its correctness is rather la-borious. The much simpler example is presented in [10]. It is an example of a topology Tf which is bigger than the density topology and invariant under
multiplication by nonzero numbers. By Theorem 23.12, such topology can not be generated by a sequence. The proof is based on the (∆2) condition which is
presented in the next chapter (compare Example 24.24).
Now we present the example from the paper [5], omitting technical details. Using this example, it can be shown that the family of topologies generated by functions from A1 is "much bigger" than the family of topologies generated by sequences.
Example 23.41([5], Theorem 6). Let us define sequences hwi := (2, 2, 3, 3, 3, 4, 4, 4, 4, . . .) , hri := (1, 2, 1, 2, 3, 1, 2, 3, 4, . . .) , a0:= 1, an:= an−1 w2 n and bn:= anwn= an−1 wn . Of course, limn→∞an−1bn = limn→∞
bn an = ∞. The function b f(x) := an−1for x ∈ (bn, an−1] , bnrn for x ∈ (an, bn] belongs to A1, and T b
f 6= Thsifor each hsi ∈ eS.
Slightly modifying the function bfone can obtain continuum different topolo-gies fromTf : f ∈ A1 \
n
Thsi: hsi ∈ eSo.
Example 23.42([7], Cor. 7 and Th. 8). Let bf be a function defined in example 23.41 and bfα(x) := bf x α for α > 1. Then bfα∈ A 1, T b fα∈/ n
Thsi: hsi ∈ eSoand T
b
fβ & Tbfα for β > α > 1.
The result from Example 23.41 can be also strengthen in a different way. Theorem 23.43 ([7, Th. 11]). Let hai , hbi ∈ eS. If Thbi& Thai then there exists
f∈ A1such thatT
hbi⊂ Tf ⊂ ThaiandTf 6= Thsifor hsi∈ eS.
23.7 f -density and ψ-density
By Theorem 23.34, it is clear that a ψ-density topology is a specific case of an f -density topology. Namely, for ψ ∈ bC, we have Tψ = T
s fψ, where fψ(x) := xψ (x). Therefore: Proposition 23.44. n Tψ : ψ ∈ bC o(1) ⊂ Tf : f ∈ A ∧ lim x→0+ f(x) x = 0 (2) ⊂Tf : f ∈ A0 .
Note that both above inclusions are proper. From Theorem 23.32, the equal-ity limx→0+ f(x)x = 0 implies Tf $ Td. On the other hand, there is f ∈ A0such
that topologies Tf and Tdare not comparable. Thus the inclusion (2) is proper.
the question whether one can replace monotonicity of ψ by monotonicity of xψ (x), in the definition of ψ-density point. The negative answer was obtained in [15]. The proof is based on the condition which is presented in the next chapter (Theorem 24.2). Theorem 23.45 ([15], Theorem 1.3). n Tψ : ψ ∈ bC o $ Tf : f ∈ Ac∧ lim x→0+ f(x) x = 0 .
The familyTf : f ∈ A0 is much bigger than the family of all ψ-density
topologies and it contains topologies incomparable with the density topology. Nevertheless, the properties of f -density topologies generated by functions from A0are very similar to the properties of ψ-density topologies. These top-ics are more precisely desribed in the next chapter.
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TOMASZFILIPCZAK
Institute of Mathematics, Łód´z Technical University ul. Wólcza´nska 215, 93-005 Łód´z, Poland
E-mail: tfil@math.uni.lodz.pl
