@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 4, No. 1, 2003

pp. 35–46

On ϕ1,2-countable compactness and filters

T. H. Yalvaç
∗

Abstract. In this work the author investigates some relations
between ϕ1,2-countable compactness, filters, sequences and ϕ1,2-closure
operators.

2000 AMS Classification: 54A20, 54D30.
Keywords: Countable compactness, filter, convergence, operation, unifica-

tion.

1. Introduction

Many generalizations of the notion of compact space have been defined in the
literature, including those of quasi H-closed space, S-closed space, rs-compact
space, feebly compact space, countably S-closed space, countably rs-compact
space, and many more. Some of these concepts have been characterized in terms
of filters and nets, and this has lead to such notions as r-convergence, RC-
convergence, SR-convergence, r-accumulation point, RC-accumulation point
and SR-accumulation point of filters and filterbases.

The notion of an operation on a topological space is a useful tool when
attempting to unify such concepts, and in earlier studies we have defined
ϕ1,2-countably compact sets, ϕ1,2-convergence of a filter and ϕ1,2-accumulation
points of a filter, and used these to obtain some such unifications.

In the present work we will study the relations between ϕ1,2-countable com-
pactness, filters, sequences and ϕ1,2-closure operators.

There are several different definitions of an operation in the literature. We
have used the one first given in [12] for fuzzy topological spaces.

In a topological space (X,τ), int , cl , scl , pcl etc. will stand for the interior,
closure, semi-closure, pre-closure operations, and so on. For a subset A of X,
Ao, Ā will also be used to denote the interior and closure of A, respectively.

∗Dedicated to the memory of Professor Doğan Çoker.



36 T. H. Yalvaç

Definition 1.1. Let (X,τ) be a topological space. A mapping ϕ : P(X) →
P(X) is called an operation on (X,τ) if ϕ(∅) = ∅ and Ao ⊆ ϕ(A), ∀A ∈ P(X).

The class of all operations on a topological space (X,τ) will be denoted by
O(X,τ).

For ϕ1,ϕ2 ∈ O(X,τ) we set ϕ1 ≤ ϕ2 ⇐⇒ ϕ1(A) ⊆ ϕ2(A), ∀A ∈ P(X).
The operations ϕ, ϕ̃ are dual if ϕ̃(A) = X \ϕ(X \A), ∀A ∈ P(X).
An operation ϕ ∈ O(X,τ) is called monotonous if ϕ(A) ⊆ ϕ(B) whenever

A ⊆ B (A,B ∈ P(X)).

Definition 1.2. Let ϕ ∈ O(X,τ). Then A ⊆ X is called ϕ-open if A ⊆ ϕ(A).
Dually, B ⊆ X is called ϕ-closed if X \B is ϕ-open.

Clearly, X and ∅ are both ϕ-open and ϕ-closed, while each open set is a
ϕ-open set for any ϕ ∈ O(X,τ).

If (X,τ) is a topological space, ϕ ∈ O(X,τ), then ϕO(X), ϕC(X) will
denote respectively the set of ϕ-open, ϕ-closed subsets of X. For x ∈ X we set
ϕO(X,x) = {U ∈ ϕO(X) | x ∈ U}.

For ϕ2,ϕ1 ∈ ϕO(X) sufficient, generally not necessarily, conditions for
ϕ1O(X) ⊆ ϕ2O(X) are ϕ2 ≥ ϕ1 or ϕ2 ≥ ı [21]. Here ı is the identity op-
eration.

Definition 1.3. For the operations ϕ1, ϕ2 ∈ O(X,τ), ϕ2 is called regular with
respect to ϕ1O(X) if for each x ∈ X and U,V ∈ ϕ1O(X,x), there exists a
W ∈ ϕ1O(X,x) such that ϕ2(W) ⊆ ϕ2(U) ∩ϕ2(V ).

Clearly, if ϕ1O(X) is closed under finite intersection and ϕ2 is monotonous,
then ϕ2 is regular w.r.t. ϕ1O(X).

Definition 1.4. Let ϕ1,ϕ2 ∈ O(X,τ), A ⊆ X, x ∈ X. Then:
(a) x ∈ ϕ1,2int A iff there exists a U ∈ ϕ1O(X,x) such that ϕ2(U) ⊆ A.
(b) x ∈ ϕ1,2cl A ⇐⇒ ϕ2(U) ∩A 6= ∅ for each U ∈ ϕ1O(X,x).
(c) A is ϕ1,2-open ⇐⇒ A ⊆ ϕ1,2int A.
(d) A is ϕ1,2-closed ⇐⇒ ϕ1,2cl A ⊆ A.

For any set A we have X \ϕ1,2int A = ϕ1,2cl (X \A) and A is ϕ1,2-open iff
X \A is ϕ1,2-closed.

Definition 1.5. [1] A subfamily U of the power set of a non-empty set X is
called a supratopology on X if ∅,X ∈U and U is closed under arbitrary unions.

If U is a supratopology on X, then the pair (X,U) is called a supratopological
space.

The notions of base, first and second countablility for a supratopology may
be defined as for topological spaces [2].

If the operation ϕ ∈ O(X,τ) is monotonous, then ϕO(X) is a supratopology.

Theorem 1.6. [22] Let ϕ1,ϕ2 ∈ O(X,τ). Then:



On ϕ1,2-countable compactness 37

(a) ϕ1,2O(X), the family of all ϕ1,2-open subsets of X, is a supratopology
on X.

(b) If ϕ2 is regular w.r.t. ϕ1O(X), then the operator ϕ1,2cl defines the
topology τϕ1,2 = {T | T ⊆ X, ϕ1,2cl (X \T) ⊆ X \T} = ϕ1,2O(X).

(c) If ϕ2 is regular w.r.t. ϕ1O(X) and ϕ1O(X) ⊆ ϕ2O(X), then the op-
erator ϕ1,2cl defines the topology τϕ1,2 = {T | T ⊆ X,ϕ1,2cl (X \T) =
X \T} = ϕ1,2O(X).

(d) If ϕ2 is regular w.r.t. ϕ1O(X), ϕ1O(X) ⊆ ϕ2O(X), and ϕ2(U) ∈
ϕ1,2O(X) for each U ∈ ϕ1O(X), then the operator ϕ1,2cl is a Kura-
towski closure operator and ϕ1,2cl A = τϕ1,2 cl A, ∀A ⊆ X.

Clearly if ϕ1 ∈ O(X,τ) is monotonous and ϕ2 = ı then ϕ1,2O(X) = ϕ1O(X)
and ϕ1,2C(X) = ϕ1C(X).

The following example illustrates the wide range of well known concepts
covered by the notions defined above.

Example 1.7. For the operations
ϕ1 = int , ϕ2 = cl ◦ int , ϕ3 = cl , ϕ4 = scl , ϕ5 = ı, ϕ6 = int ◦ cl ,
defined on a topological space we have:
• ϕ1 ≤ ϕ2 ≤ ϕ3 and ϕ1 ≤ ϕ6 ≤ ϕ4 ≤ ϕ3.
• ϕ1O(X) = τ,
• ϕ2O(X) = SO(X) = the family of semi-open sets.
• ϕ3O(X) = ϕ5O(X) = P(X) = the power set of X.
• ϕ6 = PO(X) = the family of pre-open sets.
• ϕ1,3O(X) = τθ = the topology of all θ-open sets.
• ϕ2,4O(X) = SθO(X) = the family of semi-θ-open sets.
• ϕ1,6O(X) = τs = the semi regularization topology of X.
• ϕ2,3O(X) = θSO(X) = the family of all θ-semi-open sets.
• The operations ϕ1,ϕ3 and ϕ2,ϕ6 are dual to one another.

All these operations are regular w.r.t. ϕ1O(X).

2. ϕ1,2–Countable Compactness

Definition 2.1. [21] Let ϕ1,ϕ2 ∈ O(X,τ), X ∈ A ⊆ P(X) and A ∈ P(X).
Then:

(a) If each countable A-cover U of A has a finite subfamily U′ such that
A ⊆

⋃
{ϕ2(U) | U ∈ U′}, then we say that A is (A - ϕ2)-countably

compact relative to X (for short, a (A - ϕ2)-C.C. set).
(b) We call a (A - ı)-C.C. set a A-C.C. set.
(c) If we take A = ϕ1O(X) in (a) we say that A is a ϕ1,2-C.C. set.

If we take A = ϕ1,2O(X) in (b) we say that A is a ϕ1,2O(X)-C.C. set.
If X is ϕ1,2-C.C. (ϕ1,2O(X)-C.C.) relative to itself, then X will be
called a ϕ1,2-C.C. (ϕ1,2O(X)-C.C.) space.

We remark that the condition X ∈ A is added here, and in our earlier
papers, to guarantee the existence of an A-cover or of a countable A-cover of
a subset of X. However, all the results still hold without this condition.



38 T. H. Yalvaç

One may define ϕ1,2-compact, A-compact, ϕ1,2-Lindelöf and A-Lindelöf sets
in a similar way [20, 23].

We assume that all the operations ϕi, i = 1, 2, . . . are defined on (X,τ)
whenever they are used.

Example 2.2. Let A ⊆ X.
(1) If ϕ1 = int , ϕ2 = ı, then A is a ϕ1,2-C.C. set iff A is countably compact.
(2) If ϕ1 = int , ϕ2 = cl , then A is a ϕ1,2-C.C. set iff A is feebly compact

relative to X [16], and X is ϕ1,2-C.C. iff X is feebly compact (or,
equivalently, lightly compact). X is H(1)-closed [16] iff it is a Hausdorff
first countable ϕ1,2-C.C. space with respect to these operations.

(3) If ϕ1 = cl ◦int , ϕ2 = cl , then X is ϕ1,2-C.C. iff it is countably S-closed
[6].

(4) If ϕ1 = int , ϕ2 = int ◦ cl , then X is strongly H(1)-closed [19] iff it is
a Hausdorff first countable ϕ1,2-C.C. space.

(5) If ϕ1 = cl ◦ int , ϕ2 = scl , then X is ϕ1,2-C.C. iff it is countably
rs-compact [7].

(6) For ϕ1 = int ◦ cl ◦ int , ϕ2 = ı, we have ϕ1O(X) = ϕ1,2O(X) =
τα. Hence, X is countably α-compact [13] iff it is ϕ1,2-C.C. iff it is
ϕ1,2O(X)-C.C. iff it is ϕ1O(X)-C.C.

Definition 2.3. Let F be a filter (or filterbase) on X, (xn) a sequence in X
and a ∈ X. We say that:

(a) F, ϕ1,2-accumulates to a, if a ∈
⋂
{ϕ1,2cl F | F ∈F} [20].

(b) F, ϕ1,2-converges to a, if for each U ∈ ϕ1O(X,a), there exists F ∈ F
such that F ⊆ ϕ2(U) [20].

(c) (xn), ϕ1,2-accumulates to a, if for each U ∈ ϕ1O(X,a) and for each n,
there exists an n0 such that n0 ≥ n and xn0 ∈ ϕ2(U).

(d) (xn), ϕ1,2-converges to a, if for each U ∈ ϕ1O(X,a), there exists an n0
such that for each n (n ≥ n0), xn ∈ ϕ2(U).

Example 2.4. Let F be a filter (or filterbase) on X and a ∈ X.
(1) If ϕ1 = int , ϕ2 = ı, then F, ϕ1,2-converges to a iff F converges to a in

(X,τ) and F, ϕ1,2-accumulates to a iff F accumulates to a (or a is an
adherent point of F) in (X,τ).

(2) If ϕ1 = int , ϕ2 = cl , then F, ϕ1,2-converges to a iff F, r-converges
[10] (or equivalently Θ-converges [9], almost converges [3]) to a, and
F, ϕ1,2-accumulates to a iff a is an r-accumulation point [10] (or an
almost adherent point [3]) of F.

(3) For ϕ1 = cl ◦ int , ϕ2 = cl , it can be seen that, F, ϕ1,2-converges
(ϕ1,2-accumulates) to a iff F, rc-converges (rc-accumulates) to a [9],
since {V̄ | V ∈ τ, a ∈ V̄} = {Ū | U ∈ SO(X), a ∈ U}. At the
same time, F, ϕ1,2-converges (ϕ1,2-accumulates) to a iff F, s-converges
(s-accumulates) to a [4].

(4) If ϕ1 = int ◦cl ◦int , ϕ2 = ı, then F, ϕ1,2-converges (ϕ1,2-accumulates)
to a iff F, α-converges (α-accumulates) to a [14].



On ϕ1,2-countable compactness 39

(5) If ϕ1 = cl ◦ int , ϕ2 = scl , it can be easily seen that F, ϕ1,2-converges
(ϕ1,2-accumulates) to a iff F, SR-converges (SR-accumulates) to a [5].

(6) For ϕ1 = cl ◦ int , ϕ2 = int ◦ scl , then we see that F, ϕ1,2-converges
(ϕ1,2-accumulates) to a iff F, RS-converges (RS-accumulates) to a [15].

(7) If ϕ1 = int , ϕ2 = int ◦cl then F, ϕ1,2-converges (ϕ1,2-accumulates) to
a iff F, δ-converges (δ-accumulates) to a [19].

Similar characterizations of the various notions of convergence and accumu-
lation point for sequences and nets given in the literature can be easily given,
and we omit the details.

Theorem 2.5. Let A ⊆ X and F = {Fn | n ∈ N} be a countable filterbase
which meets A. If some sequence satisfying xn ∈ (

⋂n
i=1 Fi) ∩ A for each n,

ϕ1,2-accumulates to some point a ∈ X, then the filterbase F, ϕ1,2-accumulates
to a.

Conversely if for any sequence (xn) in A the countable filterbase F = {{xm |
m ≥ n} | n ∈ N} which consists of the tails of the sequence (xn), ϕ1,2-
accumulates to some point a ∈ X, then the sequence (xn), ϕ1,2-accumulates
to a.

Proof. Let F = {Fn | n ∈ N} be a countable filterbase which meets A. Then
F′ = {

⋂n
i=1 Fi | n ∈ N} is a decreasing countable filterbase which meets A

and generates the same filter as F. Take xn ∈ (
⋂n
i=1 Fi) ∩ A for each n, and

let (xn), ϕ1,2-accumulate to a. Then, for each U ∈ ϕ1O(X,a) and for each n,
∅ 6= ϕ2(U) ∩ (

⋂n
i=1 Fi) ∩A ⊆ ϕ2(U) ∩ (

⋂n
i=1 Fi), hence ϕ2(U) ∩Fn 6= ∅. So,

F, ϕ1,2-accumulates to a.
Conversely let (xn) be a sequence in A, and let F = {Tn | n ∈ N} be the

countable filterbase consisting of the tails of (xn), which ϕ1,2-accumulate to
some point a and meets A. Then for each U ∈ ϕ1O(X,a) and for each n,
ϕ2(U) ∩Tn 6= ∅. This means that a is a ϕ1,2-accumulation point of (xn). �

Corollary 2.6. Let A ⊆ X. Each countable filterbase which meets A, ϕ1,2-
accumulates to some point of A iff each sequence in A, ϕ1,2-accumulates to
some point of A.

Theorem 2.7. Let A ⊆ X. If each countable filterbase which meets A, ϕ1,2-
accumulates to some point of A, then A is a ϕ1,2-C.C. set.

Proof. Let A ⊆
⋃
U, U = {Un | n ∈ I}, I countable and Un ∈ ϕ1O(X).

Assume that for each finite subset J of I we have A 6⊆
⋃
i∈J ϕ2(Ui). Then

A∩(X\
⋃
i∈J ϕ2(Ui)) 6= ∅. The family F = {X\

⋃
i∈J ϕ2(Ui) | J ⊆ I, J finite}

is a countable filterbase which meets A. So, A∩ (
⋂
{ϕ1,2cl F | F ∈ F}) 6= ∅.

Let F, ϕ1,2-accumulate to a ∈ A. There exists an i0 ∈ I such that a ∈ Ui0 . Now
X \ϕ2(Ui0 ) ∈ F, ϕ2(Ui0 ) ∩ (X \ϕ2(Ui0 )) 6= ∅. This contradiction completes
the proof. �

However, the converse of the above theorem need not hold. For operations
ϕ1 = int , ϕ2 = cl in (X,τ), each countable filterbase ϕ1,2-accumulates in



40 T. H. Yalvaç

(X,τ) iff (X,τ) is SQ-closed [18]. Also, (X,τ) is ϕ1,2-C.C. iff it is a feebly
compact space. Herrington [11] gave an example, occurring in [8], of a regular,
feebly compact but not countably compact space. Since this space is regular,
a ϕ1,2-accumulation point is the same as an accumulation point of a sequence
(filterbase) in (X,τ), so there is a sequence (countable filterbase) which does
not ϕ1,2-accumulate to any point in X.

Clearly any ϕ1,2-compact set is a ϕ1,2-Lindelöf set and a ϕ1,2-C.C. set. A
set is a ϕ1,2O(X)-compact set iff it is a ϕ1,2O(X)-Lindelöf set and a ϕ1,2O(X)-
C.C. set. If ϕ1,2O(X) has a countable base then each ϕ1,2O(X)-C.C. set is a
ϕ1,2O(X)-compact set.

We will define conditions (∗) and (∗∗) on the operations ϕ1 and ϕ2 in the
following way:

(∗) ϕ2 ≥ ϕ1 or ϕ2 ≥ ı,
(∗∗) ϕ2(U) ∈ ϕ1O(X) and ϕ2(ϕ2(U)) ⊆ ϕ2(U), for each U ∈ ϕ1O(X).

Example 2.8.
(1) If ϕ1 = int , ϕ2 = cl , then the condition (∗) is satisfied.
(2) If ϕ1 = cl ◦int , ϕ2 = scl , then the conditions (∗) and (∗∗) are satisfied.
(3) If ϕ1 = int , ϕ2 = int ◦cl , then the conditions (∗) and (∗∗) are satisfied.
(4) If ϕ1 = cl ◦ int , ϕ2 = cl , then the conditions (∗) and (∗∗) are satisfied.

If the condition (∗∗) is satisfied then a set is ϕ1,2-compact set iff it is both a
ϕ1,2-Lindelöf set and a ϕ1,2-C.C. set.

Theorem 2.9. Let ϕ1 be monotonous, (X,ϕ1O(X)) be a second countable
supratopological space and A ⊆ X. If A is a ϕ1,2-C.C. set then each filterbase
which meets A, ϕ1,2-accumulates to some point of A.

Proof. Let the supratopology ϕ1O(X) have a countable base, A be a ϕ1,2-C.C.
set and F a filterbase which meets A.

Assume that A∩ (
⋂
{ϕ1,2cl F | F ∈F}) = ∅. For any x ∈ A, there exists a

Ux ∈ ϕ1O(X,x) and an Fx ∈ F such that ϕ2(Ux) ∩Fx = ∅. Now, U = {Ux |
x ∈ A} is a ϕ1-open open cover of A. Since ϕ1O(X) has a countable base, U has
a countable subfamily which covers A. Since A is a ϕ1,2-C.C. set, there exists
a finite subfamily {Ux1,Ux2, . . . ,Uxn} of U such that A ⊆

⋃n
i=1 ϕ2(Uxi). Now

(
⋃n
i=1 ϕ2(Uxi)) ∩ (

⋂n
i=1 Fxi) = ∅, so A ∩ (

⋂n
i=1 Fxi) = ∅. This contradiction

completes the proof. �

Corollary 2.10. Under the assumptions of Theorem 2.9., the following are
equivalent.

(a) A is a ϕ1,2-C.C. set.
(b) A is a ϕ1,2-compact set.
(c) Each countable filterbase which meets A, ϕ1,2-accumulates to some

point of A.

Proof. In [20], it is shown that A is a ϕ1,2-compact set iff each filterbase which
meets A, ϕ1,2-accumulates to some point of A. Since each ϕ1,2-compact set is
a ϕ1,2-C.C. set, the proof is now clear from Theorem 2.7. �



On ϕ1,2-countable compactness 41

Theorem 2.11. Let ϕ1, ϕ2 be monotonous and suppose that the conditions (∗)
and (∗∗) hold. If the supratopology ϕ1O(X) has a countable base B(ϕ1O(X)),
then B′ = {ϕ2(U) | U ∈B(ϕ1O(X))} is a countable base for the supratopology
ϕ1,2O(X).

Proof. Under the given conditions, B = {ϕ2(U) | U ∈ ϕ1O(X)} is a base for
the supratopology ϕ1,2O(X) and B′ ⊆ B ⊆ ϕ1,2O(X). Let V ∈ ϕ1,2O(X)
and x ∈ V . There exists a U ∈ ϕ1O(X,x) such that ϕ2(U) ⊆ V . Hence,
x ∈ U ⊆ ϕ2(U) ⊆ V . There exists a U

′
∈ B(ϕ1O(X)) such that x ∈ U

′
⊆ U.

Hence, we have x ∈ ϕ2(U′) ⊆ ϕ2(U) ⊆ V and ϕ2(U′) ∈B′. �

Theorem 2.12. Let (∗) and (∗∗) hold and let B = {ϕ2(U) | U ∈ ϕ1O(X)}.
Then the following are equivalent for any subset A of X.

(a) A is a ϕ1,2-compact set.
(b) A is a B-compact set.
(c) A is both a ϕ1,2-Lindelöf set and a ϕ1,2-C.C. set.
(d) A is both a B-Lindelöf set and a B-C.C. set.

Proof. Under the given conditions, A is a ϕ1,2-compact set iff it is B-compact
set [20], A is a ϕ1,2-Lindelöf set iff it is a B-Lindelöf set [23], A is a ϕ1,2-C.C. set
iff it is a B-C.C. set [22]. Hence (b) ⇐⇒ (d) is now clear, as are the others. �

Theorem 2.13. Let ϕ1, ϕ2 be monotonous and suppose that the conditions (∗)
and (∗∗) hold. If the supratopology ϕ1O(X) has a countable base B(ϕ1O(X)),
or if B = {ϕ2(U) | U ∈ ϕ1O(X)} is countable, then the following are equivalent.

(a) A is a ϕ1,2-C.C. set.
(b) A is a ϕ1,2O(X)-C.C. set.
(c) A is a B-C.C. set.
(d) A is a ϕ1,2-compact set.
(e) A is a ϕ1,2O(X)-compact set.
(f) A is a B-compact set.

Proof. Under the conditions (∗) and (∗∗), (a) ⇐⇒ (c), (b) =⇒(c) and (d) ⇐⇒
(e) ⇐⇒ (f) are given in [22] and [20] respectively. If B is a countable base
of ϕ1,2O(X), then (c) =⇒(b) is clear. In the other case, B

′
= {ϕ2(U) | U ∈

B(ϕ1O(X))} is a countable base of ϕ1,2O(X) and B
′
⊆B ⊆ ϕ1,2O(X). Hence,

a B-C.C. set will be a B
′
-C.C. set and a B

′
-C.C. set will be a ϕ1,2O(X)-C.C.

set, so we have again (c) =⇒(b). In each case (b) ⇐⇒ (e) is clear. �

Theorem 2.14. Let ϕ1 be monotonous and let a ∈ X have a countable local
base Cϕ1 (a) in the supratopological space (X,ϕ1O(X)).

(1) If ϕ2 is monotonous and regular w.r.t. ϕ1O(X), then the family F =
{ϕ2(U) | U ∈ Cϕ1 (a)} is a countable filterbase and ϕ1,2-converges to a.

(2) If ϕ1O(X) is a topology and ϕ1O(X) ⊆ ϕ2O(X), then Cϕ1 (a) is a
countable filterbase which ϕ1,2-converges to a.



42 T. H. Yalvaç

Proof. (1) For U,U′ ∈ Cϕ1 (a), a ∈ U ∩ U′ and U,U′ ∈ ϕ1O(X). Since ϕ2
is regular w.r.t. ϕ1O(X), there exists a V ∈ ϕ1O(X,a) such that ϕ2(V ) ⊆
ϕ2(U) ∩ ϕ2(U′). There exists a Vc ∈ Cϕ1 (a) such that Vc ⊆ V . Since ϕ2
is monotonous, we have ϕ2(Vc) ⊆ ϕ2(V ) ⊆ ϕ2(U) ∩ ϕ2(U′). Hence F is a
countable filterbase. Let U ∈ ϕ1O(X,a). There exists a Uc ∈ Cϕ1 (a) such that
Uc ⊆ U. ϕ2(Uc) ∈ F and, since ϕ2 is monotonous ϕ2(Uc) ⊆ ϕ2(U). So, F is
ϕ1,2-convergent to a.

(2) For U,U′ ∈ Cϕ1 (a), a ∈ U∩U′ ∈ ϕ1O(X,a). There exists a Uc ∈ Cϕ1 (a)
such that Uc ⊆ U ∩U′. Hence Cϕ1 (a) is a countable filterbase.

Now, let V ∈ ϕ1O(X,a). There exists a Vc ∈ Cϕ1 (a) such that Vc ⊆ V . Since
ϕ1O(X) ⊆ ϕ2O(X), we have Vc ⊆ V ⊆ ϕ2(V ). Hence Cϕ1 (a), ϕ1,2-converges
to a. �

Theorem 2.15. Let ϕ1, ϕ2 be monotonous, let a ∈ X have a countable local
base Cϕ1 (a) in (X,ϕ1O(X)) and also let ϕ2 be regular w.r.t. ϕ1O(X). For
A ⊆ X, a ∈ ϕ1,2cl A iff there exists a filter which contains A, has a countable
base and ϕ1,2-converges to a.

Proof. Let a ∈ ϕ1,2cl A. Then for each U ∈ ϕ1O(X,a), ϕ2(U) ∩ A 6= ∅. As
in the proof of Theorem 2.14.(1), it is easly seen that Fb = {ϕ2(V ) ∩A | V ∈
Cϕ1 (a)} is a countable filterbase. The filter F generated by Fb contains A, and
{ϕ2(V ) | V ∈ Cϕ1 (a)}⊆F. Clearly F is ϕ1,2-convergent to a.

The other part of the proof is clear from Corollary 3.4. in [20]. �

Theorem 2.16. Let ϕ1, ϕ2 be monotonous, (X,ϕ1O(X)) be a first countable
supratopological space, and define cl ∗ : P(X) −→ P(X) by cl ∗(A) = {x | there
exists a filter that contains A, has a countable base and ϕ1,2-converges to x},
for each A ∈ P(X).

(1) If ϕ2 is regular w.r.t. ϕ1O(X), then cl
∗(A) = ϕ1,2cl A for each A ∈

P(X), and cl ∗ defines the topology τ∗ = {U ⊆ X | (X\U)∗ ⊆ X\U} =
ϕ1,2O(X).

(2) If ϕ2 is regular w.r.t. ϕ1O(X) and ϕ1O(X) ⊆ ϕ2O(X), then cl ∗ defines
the topology τ∗ = {U ⊆ X | (X \U)∗ = X \U} = ϕ1,2O(X).

(3) If ϕ2 is regular w.r.t. ϕ1O(X), ϕ1O(X) ⊆ ϕ2O(X), and ϕ2(U) ∈
ϕ1,2O(X) for each U ∈ ϕ1O(X), then the operator cl ∗ is a Kura-
towski closure operator defining τ∗ = {U ⊆ X | (X \U)∗ = X \U} =
ϕ1,2O(X).

Hence, if ϕ1,ϕ2 are monotonous and (X,ϕ1O(X)) is a first countable topo-
logical space, then the ϕ1,2-closure operator and the topology τϕ1,2 = {U ⊆ X |
ϕ1,2cl (X\U) ⊆ X\U} = ϕ1,2O(X) can be defined using filters with countable
bases.

Proposition 2.17. If ϕ1O(X) ⊆ ϕ2O(X) (hence, if ϕ2 ≥ ϕ1 or ϕ2 ≥ ı), then
A ⊆ ϕ1,2cl A for each A ∈ P(X).



On ϕ1,2-countable compactness 43

Proposition 2.18. If (∗∗) holds, then ϕ2(U) ⊆ ϕ1,2int (ϕ2(U)) (i.e., ϕ2(U) ∈
ϕ1,2O(X)) for each U ∈ ϕ1O(X).

Proof. Let U ∈ ϕ1O(X) and x ∈ ϕ2(U). Then x ∈ ϕ2(U) ∈ ϕ1O(X) and
ϕ2(ϕ2(U)) ⊆ ϕ2(U). So x ∈ ϕ1,2int (ϕ2(U)). �

Corollary 2.19. (a) Under the condition (∗∗), we have, ϕ1,2cl (X \ϕ2(U)) ⊆
X \ϕ2(U) for each U ∈ ϕ1O(X).

(b) If ϕ1O(X) ⊆ ϕ2O(X) and (∗∗) holds, then ϕ1,2cl (X \ ϕ2(U)) = X \
ϕ2(U) for each U ∈ ϕ1O(X).

Remark 2.20. a) If ϕ̃2 is the dual operation of ϕ2, then {X \ ϕ2(U) | U ∈
ϕ1O(X)} = {ϕ̃2(X \U) | U ∈ ϕ1O(X)} = {ϕ̃2(K) | K ∈ ϕ1C(X)}.

b) If ϕ1 is monotonous (in which case ϕ1O(X) is a supratopology), and
ϕ2(U ∪ V ) = ϕ2(U) ∪ ϕ2(V ) for each U,V ∈ ϕ1O(X), then for each finite
subfamily {U1,U2, . . . ,Un} of ϕ1O(X),

⋃n
i=1 Ui ∈ ϕ1O(X) and ϕ2(

⋃n
i=1 Ui) =⋃n

i=1 ϕ2(Ui).

Theorem 2.21. Consider the following statements:
(i) ϕ1 is monotonous.
(ii) ϕ2 is monotonous.
(iii) ϕ2 ≥ ı or ϕ2 ≥ ϕ1 (i.e. (∗)),
(iv) ∀U ∈ ϕ1O(X), ϕ2(U) ∈ ϕ1O(X) and ϕ2(ϕ2(U)) ⊆ ϕ2(U) (i.e. (∗∗)).
(v) For each U,V ∈ ϕ1O(X), ϕ2(U ∪V ) = ϕ2(U) ∪ϕ2(V ),
(vi) ϕ̃2 is the dual of ϕ2.

and
(a) A is a ϕ1,2-C.C. set.
(b) Each countable filterbase F ⊆{X \ϕ2(U) | U ∈ ϕ1O(X)} which meets

A, ϕ1,2-accumulates to some point of A.
(c) For each countable filterbase F ⊆ {X \ ϕ2(U) | U ∈ ϕ1O(X)} which

meets A, we have A∩ (
⋂
F) 6= ∅.

(d) For each decreasing countable filterbase F ⊆{X\ϕ2(U) | U ∈ ϕ1O(X)}
which meets A, we have A∩ (

⋂
{ϕ1,2clF | F ∈F}) 6= ∅.

(e) For each decreasing countable filterbase F ⊆{X\ϕ2(U) | U ∈ ϕ1O(X)}
which meets A, we have A∩ (

⋂
F) 6= ∅.

(f) If Φ is any decreasing sequence of countable non-empty ϕ1-closed sets
such that for each F ∈ Φ, A∩ ϕ̃2(F) 6= ∅, then A∩ (

⋂
Φ) 6= ∅.

Then,
(1) (b) =⇒ (d) and (c) =⇒ (e).
(2) If (iii) holds, then (c) =⇒ (b) and (e) =⇒ (d).
(3) If (iii) and (iv) hold, then (c) ⇐⇒ (b) and (e) ⇐⇒ (d).
(4) If (iv) holds, then (a) =⇒ (c).
(5) If (i) and (v) hold, then (d) =⇒ (b) and (b) =⇒ (a).
(6) If (ii) and (vi) hold, then (a) =⇒ (f).
(7) If (i), (iii), (v) and (vi) hold, then (f) =⇒ (a).



44 T. H. Yalvaç

Proof. (1) Immediate.

2) Clear from Proposition 2.17.

(3) Clear from Corollary 2.19.

(4) Let A be a ϕ1,2-C.C. set, and F = {X \ϕ2(Ui) | i ∈ I}, Ui ∈ ϕ1O(X),
be a countable filterbase which meets A. Assume that A ∩ (

⋂
F) = ∅ and

A ⊆
⋃
i∈I ϕ2(Ui). Since, ϕ2(U) ∈ ϕ1O(X), ϕ2(ϕ2(U)) ⊆ ϕ2(U), for each U ∈

ϕ1O(X), and A is a ϕ1,2-C.C. set, there exists a finite subset J of I such that,
A ⊆

⋃
i∈J ϕ2(ϕ2(Ui)) ⊆

⋃
i∈J ϕ2(Ui). We have A ∩ (

⋂
i∈J(X \ ϕ2(Ui))) = ∅.

This contradiction completes the proof.

(5) Let F ⊆ {X \ ϕ2(U) | U ∈ ϕ1O(X)} be a countable filterbase which
meets A. Then F = {Fn | n ∈ N}, where Fn = X \ ϕ2(Un), n ∈ N and
Un ∈ ϕ1O(X). Let F ′n =

⋂n
i=1 Fi for each n. Then F

′ = {F ′n | n ∈ N} is
a decreasing countable filterbase, and F ′n =

⋂n
i=1 Fi =

⋂n
i=1(X \ ϕ2(Ui)) =

X \
⋃n
i=1 ϕ2(Ui) = X \ϕ2(

⋃n
i=1 Ui). Hence, F

′ ⊆{X \ϕ2(U) | U ∈ ϕ1O(X)}.

If we assume that (d) holds then A∩ (
⋂
{ϕ1,2cl F ′n | F ′n ∈ F′}) 6= ∅. Since

F ′n ⊆ Fn for each n, we have ϕ1,2cl F ′n ⊆ ϕ1,2cl Fn. So A∩ (
⋂
{ϕ1,2cl Fn | Fn ∈

F}) 6= ∅.

Now, let us verify that (b) =⇒ (a). Let A ⊆
⋃
U, U ⊆ ϕ1O(X) and U = {Ui |

i ∈ I} be countable. Assume that for each finite subset J of I, A 6⊆
⋃
i∈J ϕ2(Ui).

Then, A ∩ (X \
⋃
i∈J ϕ2(Ui)) 6= ∅. From our hypotheses,

⋃
i∈J Ui ∈ ϕ1O(X)

and ϕ2(
⋃
i∈J Ui) =

⋃
i∈J ϕ2(Ui). So, for each finite subset J of I, we have

A∩(X\ϕ2(
⋃
i∈J Ui)) 6= ∅. Let F = {X\ϕ2(

⋃
i∈J Ui) | J ⊆ I,J finite}. Then

F ⊆ {X \ϕ2(U) | U ∈ ϕ1O(X)} and F is a countable filterbase which meets
A. There exists an a ∈ A such that a ∈

⋂
{ϕ1,2cl F | F ∈ F} and a Ua ∈ U

such that a ∈ Ua. Now, X \ϕ2(Ua) ∈F and ϕ2(Ua)∩(X \ϕ2(Ua)) = ∅. This
contradiction completes the proof.

(6) Let Φ be a countable decreasing sequence of nonempty ϕ1-closed sets
such that for each F ∈ Φ, A∩ ϕ̃2(F) 6= ∅. Assume that A∩(

⋂
Φ) = ∅. Then,

A ⊆
⋃
{X\F | F ∈ Φ}. Since for each F ∈ Φ, X\F ∈ ϕ1O(X), and A is a ϕ1,2-

C.C. set, there exists a finite subfamily Φ′ of Φ such that A ⊆
⋃
{ϕ2(X \F) |

F ∈ Φ′}. Since ϕ2 is monotonous, A ⊆ ϕ2(
⋃
F∈Φ′(X \ F)). There exists an

F ′ ∈ Φ′ such that
⋃
F∈Φ′(X\F) = X\F

′. Then A ⊆ ϕ2(X\F ′) = X\ϕ̃2(F ′),
so A∩ ϕ̃2(F ′) = ∅. This contradiction completes the proof.

(7) Let U = {Un | n ∈ N} be a countable ϕ1-open cover of A. Assume that
for each finite subset J of N, A 6⊆

⋃
i∈J ϕ2(Ui). In this case, for each finite

subset J of N, X 6=
⋃
i∈J Ui since, otherwise, we would have A ⊆

⋃
i∈J Ui ⊆⋃

i∈J ϕ2(Ui) for a finite subset J of N.

Let Fn = X \
⋃n
i=1 Ui for each n. For each n, Fn 6= ∅, Fn ∈ ϕ1C(X) and

A∩ (X \
⋃n
i=1 ϕ2(Ui)) 6= ∅. Now



On ϕ1,2-countable compactness 45

A∩ (X \
n⋃
i=1

ϕ2(Ui)) = A∩ (X \ϕ2(
n⋃
i=1

Ui))

= A∩ (ϕ̃2(X \
n⋃
i=1

Ui))

= A∩ ϕ̃2(Fn)
6= ∅.

Hence, A∩(
⋂∞
n=1 Fn) 6= ∅. But

⋂∞
n=1 Fn = X\(

⋃∞
n=1 Un) and we obtain that

A∩ (X \
⋃∞
n=1 Un) = ∅. This contradiction completes the proof. �

Example 2.22.
(1) If ϕ1 = int , ϕ2 = cl , then ϕ̃2 = int and the conditions (i), (ii), (iii),

(v) and (vi) are satisfied.
(2) If ϕ1 = cl ◦ int , ϕ2 = scl , then conditions (i), (ii), (iii), (iv) and (vi)

are satisfied, and ϕ̃2 = semi-interior is the dual of ϕ2.
(3) If ϕ1 = cl ◦ int , ϕ2 = cl , then ϕ̃2 = int and all the conditions are

satisfied.

Many known results, see for example [6,11,17,18,19], and also many new
results, may now be obtained by choosing particular operations and combining
the above results with the unifications obtained in [20-23].

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46 T. H. Yalvaç

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Received December 2001

Revised February 2003

T. H. Yalvaç

Hacettepe University,
Faculty of Science,
Department of Mathematics,
06532 Beytepe,
Ankara,
Turkey.

E-mail address : hayal@hacettepe.edu.tr


	On 1,2-countable compactness and filters. By T.H. Yalvaç