Some comments on regular and normal bitopological spaces
Some properties of regular and normal bitopological spaces are established. The classes of sets inheriting the bitopological properties of regularity and normality are found. A theorem on a finite covering of pairwise normal spaces is proved. We also study the behavior of individual multivalued mapp...
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irk-123456789-1655482020-02-15T01:27:15Z Some comments on regular and normal bitopological spaces Dochviri, І. Короткі повідомлення Some properties of regular and normal bitopological spaces are established. The classes of sets inheriting the bitopological properties of regularity and normality are found. A theorem on a finite covering of pairwise normal spaces is proved. We also study the behavior of individual multivalued mappings, taking the axioms of bitopological regularity and normality into account. Встановлено деякі властивості регулярних i нормальних бітопологічних просторів. Знайдено класи множин, що успадковують бітопологічні властивості регулярності та нормальності. Доведено теорему про скінченне покриття попарно нормальних просторів. Також вивчено поведінку конкретних багатозначних відображень з урахуванням аксіом бітопологічної регулярності та нормальності. 2006 Article Some comments on regular and normal bitopological spaces / І. Dochviri // Український математичний журнал. — 2006. — Т. 58, № 12. — С. 1720–1724. — Бібліогр.: 13 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165548 513.83 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Dochviri, І. Some comments on regular and normal bitopological spaces Український математичний журнал |
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Some properties of regular and normal bitopological spaces are established. The classes of sets inheriting the bitopological properties of regularity and normality are found. A theorem on a finite covering of pairwise normal spaces is proved. We also study the behavior of individual multivalued mappings, taking the axioms of bitopological regularity and normality into account. |
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Article |
| author |
Dochviri, І. |
| author_facet |
Dochviri, І. |
| author_sort |
Dochviri, І. |
| title |
Some comments on regular and normal bitopological spaces |
| title_short |
Some comments on regular and normal bitopological spaces |
| title_full |
Some comments on regular and normal bitopological spaces |
| title_fullStr |
Some comments on regular and normal bitopological spaces |
| title_full_unstemmed |
Some comments on regular and normal bitopological spaces |
| title_sort |
some comments on regular and normal bitopological spaces |
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Інститут математики НАН України |
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2006 |
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Короткі повідомлення |
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http://dspace.nbuv.gov.ua/handle/123456789/165548 |
| citation_txt |
Some comments on regular and normal bitopological spaces / І. Dochviri // Український математичний журнал. — 2006. — Т. 58, № 12. — С. 1720–1724. — Бібліогр.: 13 назв. — англ. |
| series |
Український математичний журнал |
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AT dochvirií somecommentsonregularandnormalbitopologicalspaces |
| first_indexed |
2025-07-14T18:53:23Z |
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2025-07-14T18:53:23Z |
| _version_ |
1837649581379158016 |
| fulltext |
UDC 513.83
I. Dochviri (Georg. Techn. Univ., Tbilisi, Georgia)
SOME COMMENTS ON REGULAR
AND NORMAL BITOPOLOGICAL SPACES
DEQKI ZAUVAÛENNQ WODO REHULQRNYX
I NORMAL|NYX BITOPOLOHIÇNYX PROSTORIV
Some properties of regular and normal bitopological spaces are established. The classes of sets
inheriting the bitopological properties of regularity and normality are found. The theorem is proved on a
finite covering of pairwise normal spaces. Also, the behavior of individual multivalued mappings is
studied taking the axioms of bitopological regularity and normality into account.
Vstanovleno deqki vlastyvosti rehulqrnyx i normal\nyx bitopolohiçnyx prostoriv. Znajdeno
klasy mnoΩyn, wo uspadkovugt\ bitopolohiçni vlastyvosti rehulqrnosti ta normal\nosti. Do-
vedeno teoremu pro skinçenne pokryttq poparno normal\nyx prostoriv. TakoΩ vyvçeno pove-
dinku konkretnyx bahatoznaçnyx vidobraΩen\ z uraxuvannqm aksiom bitopolohiçno] rehulqrnos-
ti ta normal\nosti.
New structural provisions of a basic (enveloping) set of the same logical nature
frequently play the key role in the investigation of several mathematical objects. The
consideration of two and more structures on one and the same set makes it possible to
distinguish ordered sets of such objects and also stimulates us to study intermediate
(simultaneously depending on two or several structures) constituent elements. One of
the promising, intensively developing directions of general topology is the
bitopological space theory for which fundamental concepts have already been
constructed and which evokes much interest. Attempts to inculcate the asymmetry
principle in general topology naturally led to the concept of a bitopological space. A
bitopological space as a triple (
X, τ1
, τ2
), where X is a basic set, while τ1 and τ2
are different topologies on X, was for the first time formulated by J. C. Kelly in [1].
The topics initiated by this concept not only became the object of a systematic study in
asymmetrical topology, but were also realized in other mathematical disciplines (see,
e.g., [2, 3]).
For a fixed topological space (
X, τ
), by the methods of classical topology, we can
established, only a small part of topological properties of the subsets from family 2
X.
The investigation of a bitopological structure yields more information and thus makes
it possible to consider remaining members of the family 2
X, which previously could
not be studied in the framework of general topology.
Like in general topology, in the theory of bitopological spaces the properties of
regularity and normality are important for a complete investigation of many questions.
In this paper, we study some bitopological properties, taking into account the factors of
regularity and normality, and also consider the behavior of special multivalued
mappings on these structures.
Our discussion rests on the topological concepts which can be found in the
monograph [4], while for bitopological spaces we use [2].
For a fixed bitopological space (
X, τ1
, τ2
) and any subset A ⊂ X, we denote by
τi int A and τj cl A the interior and the closure of a set A with respect to the topologies
τi and τj , respectively. Throughout the paper it is assumed that i, j ∈ { 1; 2 }, i ≠ j.
If O ⊂ X is an open set and F ⊂ X is a closed subset in the topology τi
, then we use
the notation O ∈ τi and F ∈ co τi . The family of all τi open (briefly, i-open)
neighborhoods of any set K ⊂ X is denoted by
i
X
K∑ ( ). Sets of the class i –
– Clp (X) = τi ∩ co τi are called clopens. An induced topology τi A*( ) on a fixed
subset A ⊂ X is defined usually as τi A*( ) = A ∩ τi
.
© I. DOCHVIRI, 2006
1720 ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
SOME COMMENTS ON REGULAR AND NORMAL BITOPOLOGICAL SPACES 1721
Recall that a bitopological space (
X, τ1
, τ2
) is called to be (
i, j
)-regular if, for any
point x ∈ X and every subset x ∉ F ∈ co τi \ ∅, there exist disjoint neighborhoods
U ∈
i
X
x∑ ( ) and V ∈
j
X
F∑ ( ). While, for each pair of disjoint sets F1 ∈ co τi \ ∅
and F2 ∈ co τj \ ∅ there are disjoint O 1 ∈
j
X
F∑ ( )1 and O2 ∈
i
X
F∑ ( )2 , then (
X,
τ1
, τ2
) is called p-normal [1]. The properties of (
i, j
)-regularity and p-normality as
well as various bitopological aspects of closely adjoined asymmetric objects are
investigated in [5 – 9].
It should be noted that in our discussion it is sometimes convenient to use the
following equivalent characterizations of (
i, j
)-regularity and p-normality. A
bitopological space (
X, τ1
, τ2
) is (
i, j
)-regular iff for each point x ∈ X and any
neighborhood U ∈
i
X
x∑ ( ) there exists V ∈
j
X
F∑ ( ) such that τj cl V ⊂ U . The
property of p-normality can also be formulated as follows: a bitopological space (
X,
τ1
, τ2
) is p-normal iff for each set F ∈ co τi \ ∅ and any neighborhood U ∈
j
X
F∑ ( )
there exists G ∈
j
X
F∑ ( ) such that F ⊂ G ⊂ τi cl G ⊂ U.
The theorem below gives the types of sets which inherit the properties of (
i, j
)-
regular and p-normal bitopological structures.
Theorem 1. Let a bitopological space (
X, τ1
, τ2
) contain a nonempty subset
A ⊂ X such that: (a) A ∈ j – Clp (
X
), then the (
i, j
)-regularity of (
X , τ1
, τ2
)
implies the (
i, j
)-regularity of a subspace ( )A, ,* *τ τ1 2 ; (b) A ∈ co τ1 ∪ co τ2
, then
the p-normality of (
X, τ1
, τ2
) implies the p-normality of a subspace A, ,* *τ τ1 2( ).
Proof. (a) Let us take any point x0 and its arbitrary neighborhood
U (
x0
) ∈
i
A
x∑ ( )0 . Then there is O (
x0
) ∈
i
X
x∑ ( )0 such that U (
x0
) = A ∩ O (
x0
).
The (
i, j
)-regularity of (
X, τ1
, τ2
) implies existence of W (
x0
) ∈
i
X
x∑ ( )0 such that
W (
x0
) ⊂ τj cl W (
x0
) ⊂ O (
x0
). It is easy to show that the inclusion A ∩ W (
x0
) ⊂ A ∩
∩ τj cl W (
x0
) ⊂ A ∩ O (
x0
) = U (
x0
) is valid. Moreover, A ∩ W (
x0
) ∈
i
A
x∑ ( )0 .
Since A ∈ j – Clp (
X
), we have A ∩ τ j cl W (
x0
) = τj cl (
A ∩ W (
x0
)
). Denoting
G (
x0
) = A ∩ W (
x0
) ∈
i
A
x∑ ( )0 , we obtain G (
x0
) ⊂ τ j G x* ( )cl 0 ⊂ U (
x0
).
(b) Consider any set B ∈ τi
* \ ∅ and any neighborhood O ∈
j
A
B∑ ( ). Then there
exists M ∈
i
X
B∑ ( ) such that B ⊂ O = M ∩ A ⊂ M. From the p-normality of (
X,
τ1
, τ2
) it follows existence of V ∈
j
X
B∑ ( ) such that B ⊂ V ⊂ τj cl V ⊂ M, i. e., B ⊂
⊂ A ∩ V ⊂ τi cl V ∩ A ⊂ A ∩ M = O. Taking into account A ∈ co τi , we can write
B ⊂ A ∩ V ⊂ τi cl (
A ∩ V
) ⊂ τi cl A ∩ τi cl V = A ∩ τi cl V ⊂ O . Therefore, assuming
E = A ∩ V ∈ τi
* \ ∅, we see that the inclusion B ⊂ E ⊂ A ∩ τi cl E ⊂ O is valid.
The theorem is proved.
Note that the existence of A ∈ j – Clp (
X
) is guaranteed in bitopological spaces
(
X, τ1
, τ2
) for which (
X, τj ) is disconnected.
Using doubly open coverings, in Theorem 2 we give the structural characterization
of p-normality. It must to note that the Theorem 2 was presented without proof in
[10].
Theorem 2. Let a p-normal bitopological space (
X, τ1
, τ2
) be covered by the
family F = Ok k n∈{ } =τ τ1 2 1∩ ; . Then there exists the family
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1722 I. DOCHVIRI
Fk k n∈{ } =co coτ τ1 2 1∩ ;
such that X =
k
n
kF=1∪ and Fk ⊂ Ok
, for any k = 1; n.
Proof. For n = 1, the theorem is trivial. We will prove the validity of the result
for n ≥ 2 by the method of mathematical induction. Assume that all p-normal spaces
covered by doubly open sets, with a number of elements in this covering being less
than n, satisfy the concluding part of the theorem. Following to the principles of
induction we must consider the covering F = Ok k n∈{ } =τ τ1 2 1∩ ; . Then the sets Pn ≡
≡ X O
k
n
k\ =
−
1
1∪ ∈ co coτ τ1 2∩ and Pn ⊂ On
. Consequently, there is a nonempty set
V ∈ τi , such that Pn ⊂ V ⊂ τj cl V ⊂ On
. Then X1 ≡ X \ V ∈ co τi and, by virtue of
Theorem 1, X1 1 2, ,* *τ τ( ) is a p-normal space. Moreover, X1 = k
n
kG=
−
1
1∪ , where Gk ≡
≡ O k ∩ X 1 and k = 1 1; n − . Hence there exists a family of j-closed subsets
˜
;
Fk k n{ } = −1 1
such that X1 = k
n
kF=
−
1
1∪ ˜ and F̃k ⊂ Gk for any k = 1 1; n − . Finally, if
Fk = F̃k , where k = 1 1; n − and Fn = τj cl V, then the constructed family Fk k n{ } =1;
satisfies Theorem 2.
Below, we use some special types of the multivalued maps and consider their
behaviors by the bitopological regularity and normality. For our further investigations
recall some special notions on multivalued maps. Say that a map F : (
X, τ1
, τ2
) → (
Y,
γ1
, γ2
) is multivalues if it has cardinality | F
(
x
) | > 1, at any point x ∈ X. An opposite
to multivalued map F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) usually denote by F ′ : (
Y, γ1
, γ2
) →
→ (
X, τ1
, τ2
) and define as F ′
( y
) = x X y F x∈ ∈{ }( ) , for each y ∈ Y. It is clear,
that (
F ′
) ′ = F . According [11], the sets F
(
A
) = x A
F x∈∪ ( ) and ′( )F B =
= x X B F x∈ ≠ ∅{ }∩ ( ) are called big image of A and big preimage of B . A
multivalued map F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) call to be: (a) (
i, j
)-weakly upper semi-
continuous (brief. w.u.s.c.) if for every U ∈ γi \ ∅ there exists such V ∈ τi \ ∅ that
F
(
V
) ⊂ τj cl U; (b) (
i, j
) – ∆ continuous at x0 ∈ X , if for any O ∈
j
Y
F x∑ ( )( )0
exists U ∈
i
X
x∑ ( )0 , such that F
(
U
) ⊂ O; (c) i-pointly closed if F
(
x0
) ∈ co γi
, for
any x0 ∈ X; (d) (
i, j
)-upper almost continuous (brief. u.a.c.) at x 0 ∈ X if for any
V ∈
i
Y
F x∑ ( )( )0 exists U ∈
i
X
x∑ ( )0 such that F
(
U
) ⊂ γi int γj cl V. From (b), it is
obvious what we mean under i-continuity of F.
Moreover, in the Theorems 4, 5 and 6 we need the following important special sets:
Ti
(
F, y
) = { x ∈ X | for every U ∈
i
X
x∑ ( ) there exists a point ξ ∈ U such that
y ∈ γi cl F
(
ξ
) }, for a multivalued map F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) and any point
y ∈ Y; suppose that are given the single valued map f : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) and
multivalued map F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
), then we put Ki
( f, F
) =
= ( , ) ( ) ( )a b X X f a F bi∈ × ∈{ }γ cl ; also, it may be associate to the every pair of
multivalued maps F , Φ : ( )X, ,τ τ1 2 → ( )Y, ,γ γ1 2 the set S
X Y
F
2 ,
,Φ ≡
≡ ( , ) ( ) ( )x x X X F x xi j1 2 1 2∈ × ( ) ( ) = ∅{ }γ γcl cl∩ Φ . Taking into account these
notations we obtain following results.
In the Theorem 3 are established conditions under which (
i, j
)-u.a.c. imply i -
continuity of a multivalued map. Just as in [12], we say that a subset A in a
bitopological space (
X, τ1
, τ2
) is (
i, j
)-paracompact, if every i-open covering of A,
contains j-locally finite i-open refinement, which covers A.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
SOME COMMENTS ON REGULAR AND NORMAL BITOPOLOGICAL SPACES 1723
Theorem 3. Let a multivalued map F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) be (
i, j
)-u.a.c.
and F
(
x
) ⊂ Y be an (
i, j
)-paracompact subset, for each x ∈ X. If a bitopological
space (
Y, γ1
, γ2
) is (
i, j
)-regular, then F is i-continuous map.
Proof. Consider arbitrary point x0 ∈ X and any O ∈
i
Y
F x∑ ( )( )0 . Then by (
i,
j
)-regularity of (
Y, γ 1
, γ 2
) follows that for each y k ∈ F
(
x0
) there exists
Gk ∈
i
Y
ky∑ ( ), such that γj cl Gk ⊂ O. It is obvious, that F
(
x0
) ⊂
y F x k
k
G∈ ( )0
∪ ⊂
⊂
y F x j k
k
G∈ ( )0
∪ γ cl ⊂ O. Because F
(
x0
) ⊂ Y is an (
i, j
)-paracompact, then there
exists j-locally finite cover of X A A iξ ξ ξ
γ∈{ } ∈Ω
, such that Aξ ⊂ Gk for some Gk
.
Note that takes place the implications F
(
x0
) ⊂
ξ ξ∈Ω∪ A ⊂
y F x k
k
G∈ ( )0
∪ ⊂
⊂
y F x j k
k
G∈ ( )0
∪ γ cl ⊂ O and F
(
x0
) ⊂
ξ ξ∈Ω∪ A ⊂
ξ ξγ∈Ω∪ j Acl ⊂
⊂
y F x j k
k
G∈ ( )0
∪ γ cl ⊂ O . Assume that A ≡
ξ ξ∈Ω∪ A . Since Aξ ξ{ } ∈Ω
is j-
locally finite family then γj cl A = ξ ξγ∈Ω∪ j Acl and respectively have F
(
x0
) ⊂ A ⊂
⊂ γ j Acl ⊂ O. By taking into account (
i, j
)-u.a.c. of F, and the implication F
(
x0
) ⊂
⊂ A yields existence such V ∈
i
X
x∑ ( )0 that F
(
x
) ⊂ γi int γj cl A for each x ∈ V.
Therefore the implication F
(
x
) ⊂ γi int γj cl A ⊂ γj cl A ⊂ O is valid for any x ∈ V,
i. e., F is i-continuous at x0 ∈ X.
Is valid the following theorem.
Theorem 4. Let (
X, τ1
, τ2
) be a j-cofinite and (
Y, γ1
, γ2
) be an (
i, j
)-
regular bitopological space. If a multivalued map F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) is
the (
i, j
) – ∆ continuous, i-pointly closed and | F ′
( y
) | < ℵ0
, at every point
y ∈ Y. Then the set Ti
(
F, y
) ∈ co τj .
Proof. Indeed, suppose that F is an (
i, j
) – ∆ continuous and y ∉ F
(
x0
) ∈ co γi
for some x0 ∈ Ti
(
F, y
). Then by (
i, j
)-regularity of (
Y, γ1
, γ2
) implies existence of
the disjoint sets V1 ∈
i
Y
y∑ ( ) and V 2 ∈
j
Y
F x∑ ( )( )0 . Since F is (
i , j
) – ∆
continuous, then there is such U ∈
i
X
x∑ ( )0 that F
(
U
) ⊂ V2. It is obvious that, if
x ∈ U then F
(
x
) ⊂ F
(
U
) ⊂ V2 and F
(
x
) ∩ V1 = ∅, this implies x0 ∉ Ti
(
F, y
), i. e.,
Ti
(
F, y
) ⊂ F ′
( y
). Therefore | Ti
(
F, y
) | < ℵ0 and we have Ti
(
F, y
) ∈ co τj .
It is well-known that a bitopological space (
X , τ1
, τ 2
) is p-extremally
disconnected (brief. p-E.D.) iff τi cl O1 ∩ τj cl O2 = ∅, for any pair of the disjoint sets
O1 ∈ τj and O2 ∈ τi [13]. Using the notion of p-E.D., in the Theorem 5, we give a
relation between the single valued and multivalued maps.
Theorem 5. Let f : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) be a single valued (
i , j
) – ∆
continuous map and F : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
) be (
i, j
)-w.u.s.c. multivalued
map. If (
Y, γ1
, γ2
) is the (
i, j
)-regular, p-E.D. bitopological space, then the set
Kj
( f, F
) is j-closed.
Proof. Let us consider (
a, b
) ∈ X × X \ Kj
( f, F
), then f (
a
) ∉ γj cl F (
b
). By
(
i, j
)-regularity of (
Y, γ1
, γ2
) follows existence the disjoint sets V ∈
j
Y
f a∑ ( )( ) and
W ∈
i
Y
j F b∑ ( )γ cl ( ) . Obviously, from p-E.D. implies that γi cl V ∩ γ j cl W = ∅.
From (
i, j
) – ∆ continuity of f and (
i, j
)-weakly upper semicontinuity of F, follows
existence of a pair of the sets A ∈
i
X
a∑ ( ) and B ∈
i
X
B∑ ( ) , such that f (
A
) ⊂ V
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
1724 I. DOCHVIRI
and F
(
B
) ⊂ γj cl W. It is clear that A × B ∈
i
X X
a b
×∑ ( , ) and (
A × B
) ∩ Kj
( f, F
) =
= ∅, this implies that Kj
( f, F
) ∈ co (
τi × τi
).
At the end of the paper we prove the following important result.
Theorem 6. Consider the ( j, i
)-w.u.s.c. and (
i, j
)-w.u.s.c. multivalued maps F,
Φ : (
X, τ1
, τ2
) → (
Y, γ1
, γ2
), where (
Y, γ1
, γ2
) is p-normal bitopological space.
Then the set S
X Y
F
2 ,
,Φ ∈ co (
τj × τi
).
Proof. Suppose that (
x1
, x2
) ∈ X × X \ S
X Y
F
2 ,
,Φ , then γ i cl F
(
x1
) ∩ γj cl Φ
(
x2
) ≠ ∅.
Since (
Y, γ1
, γ2
) is p-normal, then there are the sets V ∈
j
Y
i F x∑ ( )γ cl ( )1 and
W ∈
i
Y
j x∑ ( )γ clΦ( )2 such that γi cl V ∩ γj cl W = ∅. By the ( j, i
)-w.u.s.c. of F
and (
i, j
)-w.u.s.c. of Φ , follows existence of the sets A ∈
j
X
x∑ ( )1 and
B ∈
i
X
x∑ ( )2 , such that F
(
A
) ⊂ γi cl V and Φ
(
B
) ⊂ γj cl W. Consequently, it takes
place A × B ∈ τ τj i
X X
x x×
×∑ ( , )1 2 and (
A × B
) ∩ S
X Y
F
2 ,
,Φ = ∅, i.e., S
X Y
F
2 ,
,Φ ∈ co (
τj × τi
).
According [5], a space (
X, τ1
, τ2
) is said to be (
i, j
)-stable if its every nonempty
i-closed subset is j-compact. Several related properties such bitopological spaces are
obtained in [6], too. Finally we note, that if in the Theorems 4, 5 and 6 the suitable
bitopological spaces are stable, then the sets T i
(
F , y
), Kj
( f , F
) and S
X Y
F
2 ,
,Φ
respectively are compact.
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Received 14.02.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 12
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