SOOCHOW JOURNAL OF MATHEMATICS
Volume 31, No. 2, pp. 143-153, April 2005
A STUDY OF APPROXIMATELY CONTINUOUS
FUNCTIONS IN A METRIC SPACE
BY
PRATULANANDA DAS AND MD. MAMUN AR. RASHID
Abstract. The notion of approximately continuous functions is introduced in a
metric space X by using the concept of density topology and several properties
of such functions are established. We then construct the space of such functions
from X to a topological space and show that this space is metrizable under certain
conditions.
1. Introduction
The notion of approximately continuous functions, which is a major source
of formulating the concept of density topology and density function, has been
widely studied in real number space (see for example [8], [9], [19]) and then in
various abstract spaces (see for example [10], [11], [13], [18]). In each of these
cases, the density function used to define the concept of approximate continuity
always lies between 0 and 1.
In this paper we consider the situation in a metric space. In a metric space,
using a specified class of subsets, Eames [7] defined the density function which
happens to exceed 1 sometimes. In fact his density function lies between 0 and
∞. Recently Lahiri and Das [12] studied the corresponding density topology.
With the help of [7] and [12], we introduce the concept of approximate continuity
in a metric space and prove some of its basic properties. As in [4] and [12] our
nature of study does not appear to be analogous to the known methods because
Received February 25, 2002; revised December 14, 2004.
AMS Subject Classification. Primary 54A05; secondary 54C08.
Key words. metric space, density topology, approximately continuous functions, K0 -topology.
143
144
PRATULANANDA DAS AND MD. MAMUN AR. RASHID
of the unboundedness of the density function. In the last section we topologise
the space of all approximately continuous functions in a different way from [13]
and show that this space is metrizable under some general situation.
2. Preliminaries
Let (X, d) be a metric space. Let C be a class of closed sets from (X, d) and
τ be a non-negative real valued set function on C. We assume that the empty
set φ and all the singleton sets are in C, finite union of members of C is in C and
that τ (I) = 0 if and only if I contains at most one point. For each A ⊂ X, let
µ∗ (A), 0 ≤ µ∗ (A) ≤ ∞ be defined by
"
µ∗ (A) = lim+ inf
→0
∞
X
#
τ (I(n)) ,
n=1
where the infimum is taken over all possible countable collections of sets I(n)
S
from C such that A ⊂ ∞
n=1 I(n) and the diameter of I(n), diam(I(n)) < for
all n. As in Eames [7] we assume that such a countable collection of sets from C
exist for each set A and every > 0. Then µ ∗ is an outer measure function ([15],
T
T
p.35). A set A is measurable if µ∗ (B) = µ∗ (A B) + µ∗ (Ac B) for every set
B ∈ C where c stands for the complement. All Borel sets of (X, d) are measurable
([15], pp.102-106). For every set A in X, there is a measurable set B called a
measurable cover for A, such that A ⊂ B and µ ∗ (A) = µ∗ (B) ([15], pp.107-108)
and so µ∗ is a regular outer measure function. Let S denote the collection of all
µ∗ -measurable sets and µ be the measure induced by µ ∗ on S.
Definition 1.([7]) Let A ⊂ X and p ∈ X. Then the number D(A, p),
0 ≤ D(A, p) ≤ ∞, called the density of A at p is defined by
D(A, p) = lim sup
→0+
µ∗ (A I)
,
τ (I)
T
where the supremum is taken over all sets I from C such that p ∈ I and diam(I) <
∗
. Also when τ (I) = 0 or ∞, we take µ τ(A∩I)
= 0.
(I)
In [7] it is proved that if the sets in C satisfy certain regularity conditions
and µ∗ (A) is finite, then
D(A, p) = 1,
for almost all p ∈ A.
(1)
A STUDY OF APPROXIMATELY CONTINUOUS FUNCTIONS
D(A, p) = 0,
for almost all p ∈ Ac if and only if A is measurable.
145
(2)
Throughout we assume that the regularity conditions as given in [7] are
satisfied so that (1) and (2) hold.
By sets we shall always mean subsets of X and in our discussions we treat
only sets having finite measure.
Definition 2.([12]) Let D = {U ⊂ X; D(X − U, x) = 0 for all x ∈ U }. Then
D is a topology on X, called the density topology and thus (X, D) is a topological
space. Sets in D are called d-open.
We shall also make use of the following theorems of Lahiri and Das (see [12],
Theorems 1, 6 and 11, respectively).
Theorem A. The set function D(·, p) for a fixed p ∈ X is monotone nondecreasing and finitely subadditive.
Theorem B. If E is measurable then the set {x ∈ E, D(X − E, x) = 0} is
the d-interior of E.
Theorem C. Compact sets in (X, D) are finite.
We now consider the measure space (X, S, µ), where S, µ are as defined
above.
A class β ⊂ S is said to cover indefinitely a set A ⊂ X if for each x ∈ A, there
is a sequence {An } ∈ β such that x ∈ An for all n and µ(An ) → 0 as n → ∞.
A family A ⊂ S is called µ-regular ([11], for further details see [1], [5]) if
(i) µ∗ (
S
A∈A A)
< ∞,
(ii) the set ρ(A) of points outside A and indefinitely covered by subsets A 0
disjoint to A has measure zero,
(iii) there exist numbers a, b (b > a > 1) such that, for every A ∈ A, µ ∗ (Ω(A)) <
b µ(A), where Ω(A) = ∩{A0 ; A0 ∈ A, A ∩ A0 6= ∅, µ(A0 ) < a µ(A)}.
The following theorem, which we refer here as the Vitali Theorem can also
be proved there as in ([5]; see also [1]).
Vitali Theorem.([1], Theorem 1) Let A ∈ S be a set indefinitely covered by
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PRATULANANDA DAS AND MD. MAMUN AR. RASHID
a regular family A. Then there is a sequence {A n } of disjoint sets in A such that
"
∞
[
µ A−A∩
An
n=1
!#
= 0.
Also, for every > 0, there is a sequence {A n } of disjoint sets in A such that
µ
"
∞
[
#
An < µ(A) + .
n=1
We shall need the following application of the Vitali Theorem due to Lahiri
and Chakraborty [11] to investigate the separability of the range of an approximately continuous function (Theorem 5).
Corollary A. Let > 0 be arbitrary. Under the conditions of the Vitali
Theorem, there is a positive integer k depending on such that
"
µ A−
k
[
r=1
#
Ar < .
3. Approximately Continuous Functions
Let (Y, τ ) be a topological space. By mappings, we shall always mean mappings (functions) from X into Y , unless otherwise mentioned. Further we shall
assume the definition of measurability of functions as in [6].
Definition 3.(Definition 1, [11]) A mapping f is said to be approximately
continuous at α ∈ X if for every G ∈ τ with f (α) ∈ G, there is a set E ∈ S with
D (X − E, α) = 0 such that f (E) ⊂ G.
If f is approximately continuous at each α ∈ X, then f is called an approximately continuous function.
Definition 4.([3], [11], [16], [17]) A mapping f is said to be continuous with
respect to the density topology at x ∈ X if, for every G ∈ τ with f (x) ∈ G, there
is a d-open set E with x ∈ E and f (E) ⊂ G.
If f is continuous with respect to the density topology at each x ∈ X, then
f is continuous with respect to the density topology.
A STUDY OF APPROXIMATELY CONTINUOUS FUNCTIONS
147
Theorem 1. A mapping f is approximately continuous if and only if it is
continuous with respect to the density topology.
Proof. We first suppose that f is approximately continuous. Let x ∈ X and
G ∈ τ be such that f (x) ∈ G. Then there is a set E ∈ S with D(X − E, x) =
S
0 and f (E) ⊂ G. Take T = E {x}. Then T is obviously measurable and
D(X − T, x) ≤ D(X − E, x) = 0 i.e. D(X − T, x) = 0. Also f (T ) ⊂ G. Let
H = {y ∈ T ; D(X − T, y) = 0}. Then by Theorem B, H being the d-interior of T
is a d-open set containing x and f (H) ⊂ f (T ) ⊂ G. Hence f is continuous with
respect to the density topology.
Conversely let f be continuous with respect to the density topology. Let
x ∈ X and G be a τ -open set containing f (x). Then there is a d-open set
E containing x with f (E) ⊂ G. Then in view of Definition 1 and (2), E is
measurable and also D(X − E, x) = 0. So f is approximately continuous.
The following results now follow immediately.
Corollary 1. Any approximately continuous function f : X → Y has the
property of Baire.
In the following three theorems we shall study the relation between the approximate continuity and measurability of a function.
Note 1. It follows from Theorem 1, that if f is approximately continuous,
then it is measurable, for if G ∈ τ , then f −1 (G) is d-open and so f −1 (G) ∈ S in
view of (2). However a stronger result is proved in the following theorem.
Theorem 2. If f is approximately continuous at almost all points of X,
then it is measurable.
Proof. Let G ∈ τ and E = f −1 (G). If µ∗ (E) = 0, then obviously E is
measurable. Suppose that µ∗ (E) > 0. Let y ∈ E be a point of approximate
continuity. Then there is a E0 ∈ S with D(X − E0 , y) = 0 and f (E0 ) ⊂ G
i.e. E0 ⊂ E. Then
D(X − E, y) ≤ D(X − E0 , y) = 0.
This is true for almost all points of E. Therefore by (2) X − E ∈ S and so E ∈ S.
So f is measurable.
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PRATULANANDA DAS AND MD. MAMUN AR. RASHID
Theorem 3. If f is measurable and (Y, τ ) is second countable, then f is
approximately continuous at almost all points of X.
Proof. Let {Un } be a countable base of τ . Then Mn = f −1 (Un ) ∈ S for all
n. For each n, let An = {x ∈ Mn ; D(X −Mn , x) = 0}. Let Tn = Mn −An . Now by
(2), Tn is measurable and µ(Tn ) = 0. Now D(X −An , y) = D[(X −Mn )
S
Tn , y] ≤
D(X − Mn , y) + D(Tn , y) = 0 + 0 = 0 for all y ∈ An (by Theorem A). Let
T =
S∞
n=1 Tn .
Then T is measurable and µ(T ) = 0. Let x ∈ X − T and U ∈ τ be
such that f (x) ∈ U . Then there is a positive integer k such that f (x) ∈ U k ⊂ U .
So x ∈ f −1 (Uk ) = Mk . Since x 6∈ T , x ∈ Ak . Further f (Ak ) ⊂ f (Mk ) ⊂ Uk .
This shows that Ak is a measurable set with D(X − Ak , x) = 0 and f (Ak ) ⊂ U .
Hence f is approximately continuous at x. This proves the theorem.
Combining Theorems 2 and 3 we have
Theorem 4. If (Y, τ ) is second countable, then a mapping f is measurable
if and only if it is approximately continuous at almost all points of X.
In the following theorem we prove that the range of an approximately continuous function has a special property under certain conditions (cf. [8], [11]).
Theorem 5. Let (Y, ρ) be a metric space and f : X → Y be approximately
continuous. Then f (X − M ) is a separable subspace of Y , where µ(M ) = 0,
provided the set function τ is bounded and finitely additive and C is µ-regular.
Proof. Let α ∈ X and > 0 be arbitrary. Then there is a set E ∈ S such
that D(X − E, α) = 0 and ρ(f (x), f (α)) < for all x ∈ E.
Now since D(X −E, α) = 0, if we write an = supdiam(In )< 1
n
µ[(X−E)∩In ]
τ (In )
where
In ∈ C and α ∈ In for all n, then an → 0 as n → ∞. Again for each n ∈ N , we
(α)
can find an In
(α)
∈ C containing α with diam(In ) < 1/n such that
an ≥
(α)
h
µ (X − E)
τ
T
(α)
In
(α)
In
i
> an −
1
.
n2
(3)
(α)
Hence {In } is a sequence of closed sets from C, all containing α with diam(I n )
A STUDY OF APPROXIMATELY CONTINUOUS FUNCTIONS
149
< 1/n such that
h
µ (X − E)
lim
n→∞
T
(α)
In
(α)
τ (In )
i
= 0.
(From (3))
Thus we can find a positive integer m such that
h
µ (X − E)
T
(α)
In
(α)
τ (In )
i
< ∀n ≥ m.
(α)
Thus for all large n (precisely n ≥ m), ρ(f (x), f (α)) < for all points x in I n
(α)
except for a subset of In
(α)
whose measure is less than τ (In ).
Now such a sequence exists for each α ∈ X and they form an indefinite
(x )
cover of X. Hence by Corollary A, there exists a finite disjoint family {I ni i },
i = 1, 2, . . . , k such that
"
µ X−
k
[
i=1
#
In(xi i ) < .
(x )
The set Ini i has the property that for all y in that set except for a subset of
(x )
measure less than τ (Ini i ) we have ρ(f (xi ), f (y)) < .
Let A = {f (x1 ), f (x2 ), . . . , f (xk )}. So except for a set of measure less than
"
τ (In(x11 ) ) + · · · + τ (In(xkk ) ) + µ X −
= τ
k
[
In(xi i )
i=1
!
k
[
i=1
In(xi i )
#
+ < (q + 1)
(since τ is finitely additive and boundedness of τ implies that τ (I) < q (say)
∀I ∈ C) for all points z in X, f (z) is at a distance less than from A .
Now let {v } be a sequence of numbers such that v > 0, v+1 < v and
limv→∞ v = 0. Now take A =
S∞
v=1
Av . Then A is a countable subset of f (X)
and dense in f (X − M ) where M is a subset of measure zero. This completes the
proof of the theorem.
Corollary 2. If in addition to the conditions of Theorem 5, D(X, p) > 0
∀p ∈ X, then f (X) is a separable subspace of Y .
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PRATULANANDA DAS AND MD. MAMUN AR. RASHID
Proof. We have already proved in Theorem 5, that there is a subset M of
X with µ(M ) = 0 such that f (X − M ) is separable that is f (X − M ) has a
countable dense subset A. We shall now show that under the given condition A
is actually dense in f (X). Let w ∈ M . From the approximate continuity of f ,
we obtain a E ∈ S with D(X − E, w) = 0 such that ρ(f (w), f (x)) < δ/2 ∀x ∈ E.
Now if E ⊂ M , then µ(E) = 0 and so D(E, w) = 0. Then
D(X, w) ≤ D(X − E, w) + D(E, w) = 0,
a contradiction. Hence there is a w1 ∈ X − M such that ρ(f (w), f (w1 )) < δ/2.
So there is a u ∈ A such that ρ(f (w1 ), u) < δ/2 and hence ρ(f (w), u) < δ. So A
is dense in f (X).
4. K0 -Topology
If X and Y are topological spaces, Arens [2] introduced a topology on the
set of all continuous functions from X to Y as follows. If K ⊂ X is compact
and W ⊂ Y is open then the collection of all continuous mappings f satisfying
f (K) ⊂ W is denoted by the symbol (K, W ). The collection of all such (K, W )
form a sub-base of a topology which is called by Arens a K-topology.
Following the line of Arens, Lahiri and Das [13] recently introduced the
concept of Kd -topology for the set of all approximately continuous functions
from X to Y , taking K, a d-compact subset of X.
We denote by AC the collection of all approximately continuous functions
from a metric space (X, d) to a topological space (Y, τ ). If K ⊂ X is finite and
W ⊂ Y is open then the collection of all approximately continuous functions f
satisfying f (K) ⊂ W is denoted by the symbol (K, W ). Then all such sets (K, W )
form a subbase of a topology on AC which we denote by K 0 -topology. Further
we observe that for any f ∈ AC there exists a basis of K 0 -open neighbourT
T
hoods of f of the form U (f ) = (K1 , W1 ) · · · (Kn , Wn ) where Ki are finite,
Wi open and f (Ki ) ⊂ Wi for i = 1, . . . , n. The set U (f ) will be denoted by
(K1 , . . . , Kn ; W1 , . . . , Wn ).
Note 2. Since for a metric space, d-compact sets are finite (by Theorem
C), K0 -topology defined above coincides with the K d -topology of [13]. But if we
A STUDY OF APPROXIMATELY CONTINUOUS FUNCTIONS
151
consider the situation in an arbitrary topological space as in [13], then clearly
every K0 -open set is Kd -open but the converse may not be true (because one of
the requirements for d-compact sets to be finite is that the measure concerned
must be an atomic measure (see [12], [20]) which is not assumed for the measure used to define the density topology on a topological space ([13])). Further
investigation is needed in this aspect. However it appears that the methods of
proofs for K0 -topology are somewhat different from that of [13] and moreover the
results obtained in this section are also valid for the K 0 -topology on an arbitrary
topological space. Therefore for the sake of completeness we give the proofs of
the following theorems.
In the next few theorems we shall show that the K 0 -topology is metrizable
under certain conditions (cf. [13], Theorem 9 and 10) which is different from
Arens’ treatment.
Theorem 6. The K0 -topology in AC is regular T1 if Y is so.
Proof. First let Y be a T1 space. Let f , g ∈ AC and f 6= g. Then there is
a x ∈ X such that f (x) 6= g(x). Now there are U , V ∈ τ satisfying f (x) ∈ U ,
g(x) 6∈ U , g(x) ∈ V , f (x) 6∈ V . Then we have f ∈ ({x}, U ) and g ∈ ({x}, V ) and
clearly g 6∈ ({x}, U ) and f 6∈ ({x}, V ). So K 0 -topology is T1 .
Let now Y be regular. Let f ∈ AC and U be a K 0 -open set containing
f . Then there is a set of the form U (f ) = (K 1 , . . . , Kn ; W1 , . . . , Wn ) such that
(i)
(i)
f ∈ U (f ) ⊂ U . Now let Ki = {x1 , . . . , xri }. Since for each j = 1, 2, . . . , ri ,
(i)
(i)
f (xj ) ∈ Wi , there is a Vj
(i)
(i)
f (xj ) ∈ Vj
Then f (Ki ) ⊂
(i)
j=1 Vj
Sr i
∈ τ such that
(i)
⊂ V j ⊂ Wi
(since Y is regular).
= Gi (say). Then f (Ki ) ⊂ Gi ⊂
(i)
j=1 V j
Sr i
= Gi ⊂ Wi
and this is true for i = 1, 2, . . . , n. Let V (f ) = (K 1 , . . . , Kn ; G1 , . . . , Gn ), so that
f ∈ V (f ) ⊂ U (f ). We shall now show that the K 0 -closure of V (f ) is contained in
U (f ). Let g 6∈ U (f ). Then for some ` = 1, 2, . . . , n, g 6∈ (K ` , W` ) i.e. g(x) 6∈ W`
for some x ∈ K` . Then
g(x) ∈ Y − W` ⊂ Y − G` = H`
(say).
152
PRATULANANDA DAS AND MD. MAMUN AR. RASHID
Clearly ({x}, H` ) is a K0 -open set containing g which is disjoint from V (f ).
Therefore g 6∈ K0 − cl(V (f )) which implies that K0 − cl(V (f )) ⊂ U (f ). So
K0 -topology is regular.
Theorem 7. If (X, D) is locally compact and second countable and Y is
second countable, then the K0 -topology is also so.
Proof. The basis BX of (X, D) may be supposed to consist of only those
d-open sets whose d-closures are d-compact and so in view of Theorem C, all
members of BX must be finite. Let BY be the basis of Y .
We shall show that the members of the K 0 -topology of the form
V0 = (U1 , . . . , Un ; W1 , . . . , Wn ),
where Ui ∈ BX and Wi ∈ BY for i = 1, 2, . . . , n form a basis of the K 0 -topology.
Let (K, W ) be any sub-base member of the K 0 -topology. Let K = {x1 , . . . , xm }.
Now let f ∈ (K, W ). So f (xj ) ∈ W for all j = 1, . . . , m. Then there is a W xj ∈
BY such that f (xj ) ∈ Wxj ⊂ W ∀j = 1 to m. Now since f is approximately
continuous at xj , there is a Uxj ∈ BX satisfying xj ∈ Uxj and f (Uxj ) ⊂ WXj for
j = 1, . . . , m. Hence f ∈ (Uxj , Wxj ) for j = 1, . . . , m and this shows that
f ∈ (Ux1 , . . . , Uxm ; Wx1 , . . . , Wxm ) ⊂ (K, W ).
Thus the sets of the form V0 form an open base of the K0 -topology. Since BX
and BY both are countable, from the construction of the sets of the form V 0 , it
follows that the class V0 is also countable. This proves the theorem.
Note 3. The structure of the sets V0 in Theorem 7 is simpler than that in
[2] and [13], where in addition the regularity of the domain space is also required,
which we do not need here.
Combining Theorems 6 and 7 we have:
Theorem 8. If (X, D) is locally compact, second countable, Y is regular T 1
and second countable then the K0 -topology is metrizable.
A STUDY OF APPROXIMATELY CONTINUOUS FUNCTIONS
153
Acknowledgment
We are thankful to the referee for his valuable suggestions which helped to
improve the presentation of the paper.
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Department of Mathematics, Jadavpur University, Calcutta - 700 032, India.
E-mail: [email protected]