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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 25. Выпуск 3.
УДК 517 DOI 10.22405/2226-8383-2024-25-3-37-46
О компактности сильно звездных идеалов топологических
пространств
П. Бал, Р. Дас, С. Саркар
Бал Прасенджит — доктор математики, Институт дипломированных финансовых аналитиков Индийского университета Трипура (г. Камалгхат, Индия). e-mail: balprasenjitl 77@gmail. com
Дас Ракхал — доктор математики, Институт дипломированных финансовых аналитиков Индийского университета Трипуры (г. Камалгхат, Индия). e-mail: [email protected]
Саркар Сусмита — магистр, Институт дипломированных финансовых аналитиков Индийского университета Трипуры (г. Камалгхат, Индия). e-mail: susmitamsc94 @gm,ail. com
Аннотация
В этой статье мы вводим понятие сильно звездной I-компактности и изучаем некоторые ее топологические особенности. Мы представляем некоторые свойства конечных пересечений как для I-компактных пространств, так и для сильно звездных 1-компактных пространств. Наконец, мы устанавливаем связь между счетно //¿„-компактным пространством и сильно звездным /^¿„-компактным пространством. Для того чтобы выявить разницу между различными версиями компактности, мы приводим несколько контрпримеров. Также в статье поставлены некоторые открытые проблемы.
Ключевые слова: Звездный идеал, звездный I-компакт, /^¿„-компактное пространство.
Библиография: 21 названий.
Для цитирования:
Бал, П., Дас, Р., Саркар, С. О компактности сильно звездных идеалов топологических пространств // Чебышевский сборник, 2024, т. 25, вып. 3, с. 37-46.
CHEBYSHEVSKII SBORNIK Vol. 25. No. 3.
UDC 517 DOI 10.22405/2226-8383-2024-25-3-37-46
On strongly star ideal compactness of topological spaces
P. Bal, R. Das, S. Sarkar
Bal Prasenjit — Ph.D. in Mathematics, The Institute of Chartered Financial Analysts of India University Tripura (Kamalghat, India). e-mail: balprasenjitl 77@gmail. com
Das Rakhal — Ph.D. in Mathematics, The Institute of Chartered Financial Analysts of India University Tripura (Kamalghat, India). e-mail: [email protected]
Sarkar Susmita — M.Sc. in Mathematics, The Institute of Chartered Financial Analysts of India University Tripura (Kamalghat, India). e-mail: susmitamsc94 @gmail. com
Abstract
In this article we introduce the concept of strongly star I-compactness and study some of its topological features. We represent some finite intersection like properties for both I-compact spaces and strongly star I-compact spaces. Lastly we establish a relation between the countably //¿„-compact space and the strongly star //¿„-compact space. In order to identify the difference between the different versions of compactness we represent some counter examples. And some open problems are also posed in this article.
Keywords: Star Ideal, Star I-compact, //¿„-compact space.
Bibliography: 21 titles.
For citation:
Bal, P., Das, R., Sarkar, S. 2024, "On Strongly Star Ideal compactness of Topological Spaces" , Chebyshevskii sbornik, vol. 25, no. 3, pp. 37-46.
1. Introduction
The the year 1933, the concept of ideals in topological spaces were considered by Kuratowski [15] and has been studied extensively by Vaidvanathaswamy [21] in the year 1946. An ideal I in a topological space (X, t) is a non empty family of subsets of X which satisfies the following properties
(i) X^ I,
(i) A,B e I ^ A u B e I,
(ii) A e I and B c A ^ B e I.
If I is an ideal in a topological space (X, t), then the ordered triplet (X, t, I) is called an ideal topological space (in short ideal space). If I n t = {0}, then I is called a condensed ideal or a boundary ideal [10]. Some simple ideals on a space (X,t) are {0} P(X) (power set of X) and I fin, collection of all finite subset of X The concept of compactness modulo an ideal (also called I-compactness) was first established by Newcomb [17] in the year 1967. Recently this generalization of compactness has attracted a lot of mathematicians in this field [12, 20].
On the other hand, Douwen et. al. [9] generalized compactness with the help of the star operator. In a topological space (X, t ), if M C X and U is a collection of sub sets of X then star of M with respect to U is denoted by St(M,U) and is defined as St(M,U) = {U e U : U n M = 0}. If M = {x}, we write St(x,U) instead of 5i({«},U). In recent days, this operator is being used in the study of selection principles fl, 3, 14], covering properties [2, 4, 5, 8, 6, 7, 19]. In this paper we will use the concept of ideal and star operator simultaneously to generalize the study of compactness of an ideal space.
2. Preliminaries
If A C X, A will denote closure of A. For general symbols and notation of topologv, we follow fill- * _ *
A subset A of a space (X, t ) is said to be a g-closed fl 6] set if A C ^whenever A C U e t. Every closed set is a g-closed set but converse may not be true.
Proposition 1. [11] If f : (X,T,I) ^ (Y, a) is a function, then f (I) = {f (Ii) : Ii e I} is an ideal of Y.
Proposition 2. [11] If I is an ideal of subsets of X and Y C X, then Iy = {Y n I\ : I\ e I} is an ideal of subsets of Y.
Although Newcomb [17] introduced the concept of I-compactness, Rancin [18], Hamlett and Jankovic [13] studied the concept extensively.
Definition 1. [11] A subset A of an ideal space (X,t,I) is said to be compact modulo I or I-compact subset, if for every t-open cover {Ua : a e A.} of A there exists a finite subset {Uai : i = 1, 2, 3,... k} such th at X \ lj£=i Uai e I. If X itself is a I-compact subs et, then (X, t, I) is called an I-compact space.
Definition 2. [9] A topological space (X, t) is called a strongly star compact space if for every open cover U of X, there exists a finite subset F C X such that St(F, U) = X.
3. I-compactness
Since 0 e / for every ideal / of X, every compact space is an /-compact space.
Remark 1. There exists an ideal space (X,t,I) which is I compact but, not, compact.
Let X = N,A = {1, 3, 5, ...},t = {0, X} U {X/P : p e P(A)} and I = P(A).
Clearly (X, t, I) is an ideal space. Let U be an arbitrary non-trivial open cover of X and U = {u1,u2, ...,Uk} is a finite suPset of U. Then there exist pn e P(A) such that Un = X/Pn Vn e {1, 2, 3,..., k} and U = ULi un = Ut=i(x/pn) = X/(ft= i Pn) = X/(nkn=iPn) Therefore, X/ UU' = nkn=lPn e P(A) = I.
Hence (X, t, I) is an I-compact space.
Now consider the countable open cover V = {Vn : n e N}, where Vn = X/{2n-1, 2n+1, 2n+3,...}.
If possible let V = {Vni ,Vn2 ,Vns ...,Vnk} is a finite sub cover of X. Then there exists Wmax = {ni,n2,n3, ...nk} e N.
Therefore, UV = Vnmax = X/{2rimax - 1, 2nmax + 1, 2nmax + 3,...} = X
So, V can not have finite sub cover. Hence (X, t, I) can not a compact space.
Definition 3. In an ideal space (X,t, I), a fam,ily {Ha : a e A} is said to have idealized finite intersection property if for every finite subset Aq e A, P|aeA Ha e I
Theorem 1. For an ideal space (X,t,I), following statements are equivalent:
1. Every family % of closed sets having idealized finite intersection property have P|% = 0
2. (X, t, I) is I-compact.
Proof.
(1) ^ (2)
Let condition (1) holds and U = {Ua : a e A} is an open cover of the ideal space (X,t,I) Therefore, % = {Ha = X/Ua : a e A} is a family of closed sets and
n% = D«ea Ha = flaeA(X/Ua) = x/uaeA ua = x/x = 0
Therefore bv condition (1) the familv % of closed sets must not have IFI property. Thus there exists a finite srbset Ao C A
ruch that. f|«eA Ha e I ^ n«ea0 (X/Ua) e I ^ X/\JaeAo (Ua) e I
So,{Ua : a e A0} is a finite subset of U such that X/{j(Ua) e I. №nce (X,t,I) is an /-compact space.
(2) ^ (1)
Let (X, t, I) is an /-compact space and % = {Ha : a e A} is a family of closed sets having
n% = 0
Then U = {Ua = X/Ha : a e A} is a family of open sets such that U ^aeA(X/Ha) = X/n«€a Ha = X/0 = X.
U is an cover of X. But (X, t, I) is / compact.
Therefore, there exists a finite subset A0 C A Such that X/ |JUa e I
So, {Ha : a e A0} is a finite s r bset of % and Haeao %a e I which is a contradiction to the fact that H has IFI property. Hence, P| % = 0 D
4. Strongly star I-compact space
Definition 4. In an ideal space (X, t,I) a subset B C X is said to be strongly star I-compact subset if for every t-open cover of B, there exists a finite subset M C B Such that X/St(M, U) e I. If X it, self is strongly star I-compact, then we say that (X, t, I) is a strongly star I-compact space.
Since 0 e I, every strongly star compact space is a strongly star I-compact space. But the converse may not true.
Remark 2. There exists a strongly star I-compact space which is not strongly star compact. Let X = N, 0 = {0} U {{n} : n e 2N} U {{1, 2n - 1} : n e N},I = V(2N) and t be the topology generated by ft.
Let U be an arbitrary open cover of X. Then for F = {1}, N/2N C St(F, U) ^ X/St(F, U) C C 2N ^ X/St(F, U) e I.
Therefore, (X, t, I) is a strongly star I-compact space Now, consider the basic open cover ft, then for every finite subset F C X,
St(F, ft) C F U (N/2N) = X Therefore, (X, t, I) can not be a strongly star compact space.
Proposition 3. Every I-compact space is a strongly star I-compact space.
^ Let (X, t, I) is an I-compact space and U be an arbitrary open cover of X. Then there exists a finite subset U' = {U\, U2, U3,..., Uk} C U such th at X/ U^=1 Uk e I
If we take Xi e Ui, Vi = 1,2,3,...,k then F = {x1 ,x2,x3, ...,xj.} C X is finite and Ukl=iUk C St(F,U) ^ X/St(F, U) C X/ Uki=1 Uk e I
Therefore, X/St(F, U) £ I
Hence (X, т, I) is strongly star I-compact space. But the converse of the above proposition may not be true.
Remark 3. There exists a strongly star I-compact space which is not strongly star compact. Let, X = = {$}U {2N} U {{1, 2n — 1} : n £ N} ,1 = P(2N) an d т be the topology generated by p. Clearly, for every open cover U of X, if we take F = {1}, then N/2N С St(F, U) ^ X/St(F, U) С 2N
^ X/St(F, U) £ I Therefore (X,t,I) is a strongly star I-compact.
Now consider the countable open cover U = {Un : n £ N} Where Un = 2N U {1, 3, 5,..., 2n — 1} Suppose that {Uni, Un2, Un3,..., Unk} is a finite subset of X. Then there exists a nmax such that Птах = тах{П1,П2, ...,пк}•
Therefore, Ukl=lUn, = Unmax =2N U {1, 3, 5,..., 2птах — 1} ^ X/ Ul=l Uni — {2Пт(ах + ~1, 2^т<ах + 3, 2^т<ах + 5,...} ^ X/ Ukk=l Uni £ I U can not have a finite subset U such that X/ UU £ I. Therefore (X, t, I) is not I-compact.
L
/-compact space
Strongly star /-compact space
Compact space
Strongly star compact space
Рис. 1: Relation among several variations of compactness.
/
Theorem 2. g-closed subset, of a strongly star I-com,pact space is strongly star I-compact subset.
Proof. Let B be a g-closed subset of a strongly star /-compact space (X, t, I), and let Ube a r-open cover of i.e., B C UW. But B is a g-closed. Therefore, B C UU. So, X/(UU) C X/B. _
Now, V = UU (X/B) become an open cover of X. But X is a strongly star /-compact space. Therefore, there exists a finite subset F C X Such that X/St(F, V) e I. ^ X/(St(F, U) U (X/B)) e / or X/St(F, U) e I. ^ (X/St(F, U)) n B)) e ^ or B/St(F, U) C X/St(F, U) e I. ^ (X/St(F, U)) n B)) e I ot B/St(F, U) e I. ^ B/St(F, U) e / or B/St(F, U) e I.
^ B/St(F,U) e I Therefore, B is a ^^^OTglv star /-compact subset. □
Corollary 1. Every closed subset of a strongly star I-compact space is a strongly star I-compact subset.
Theorem 3. If A and B be two strongly star I-compact subset in an ideal space (X, t, I) then A U B is also an strongly star I-compact subset.
Proof. Let U = {Ua : a e A} be an r-open cover of A u where A and B are strongly star /-compact subset in the ideal space (X,t,I).
Therefore, U is an r-open cover of A as well as for B. Thus there exists a finite sets M C A and N C £ such that A/St(M, U) = h e ^d B/St(N, U) = h e I. Therefore, A = St(M, U) u /1 and B = St(N, U) u h-^ A u B = St(M, U) u St(N, U) u (h u h) ^ A u B = St(M u N, U) u (Ii u h) ^ A u B/St(M u N, U) = (Ii u h) e I
Since M and N are finite, Hence M U N C A U B is also finite.
Hence A U B is a strongly star /-compact subset of X. □
Corollary 2. Finite union of strongly star I-compact subset is a strongly star I-compact subset.
Theorem 4. In a strongly star I-compact space (X, t,I) if B C X is a clopen subset of X, then (B,tb, Ib) is a strongly star Is-compact space.
Proof. ^Let (X,t, I) is strongly star /-compact space and B C X is cl-open. Let U be ar^ open cover of But B is open, therefore, U C r. Also B is closed. Hence V = UU {X/B} is a t open cover of X. But (X,t,I) is ^^^OTglv star /-compact. Therefore, there exists a finite subset M C X such that X/St(M, V) e I ^ B/St(M, V) e IB- We take P = M n B C B Which is a finite subset of B.
Now St(M, V) = St(P, U) U (X/B) or St(M, V) = St(P, U) In both cases B/St(P, U) = = St(M, V) e Ib Hence, (B,tb,Ib) is strongly star /^-compact space. □
Theorem 5. f : (X, t, I) ^ (Y, a) be function from a strongly star I-compact space to a topological space (Y,a) If f is continuous then f (x) is strongly star f (I) compact subset ofY.
Proof.
Let f : (X, t, I) ^ (Y, a) is a continuous function and (X, t, I) is strongly star /-compact. Let, U = {Ua : a e A} be a cover of f (X).
Therefore, V = {f-1(Ua) : a e A} is an cover of ^^ut X is strongly star /-compact. Therefore there exists a finite set F C X, such that X/St(F, V e I. ^ f (X/St(F, V)) e f (I) ^ f (X)/f (St(F, V)) C f (X/St(F, V)) e f (I) ^ f (X )/f (U{f-1(Ua) ev : F n f-1 (Ua) = 0}) e f (I) ^ f (X)/ U {Ua eu : f(F) n Ua = 0} e f(I) ^ f (X/St(F, U)) e f (I)
Here, If (F)| < IF I, ^teefore f (F) is finite. Hence, f (X) is a ^^^OTglv star f (Z)-compact space. □
5. Modified and idealized finite intersection property:
In an ideal space (X, t,I), a familv % of subsets of X is said to have MIFIP if for every finite subset P C X, if n{H e % : P n (X/H) = 0} / I
Theorem 6. In an ideal space (X,t,I), following statements are equivalent:
1. Every family of closed sets having MIFIP has non empty intersection.
2. (X, t, I) is strongly star I-compact space.
Proof. (1) ^ (2)
Let condition 1 holds and U = {Ua : a e A} is an open cover of X. Then % = [X/Ua : a e A} is a family of closed sets such that n% = X/{jaea Ua = X/X = 0.
Since, intersection is empty. The family % Must not have MIFIP. Therefore there exists a finite set P C X such that
0{H e % : P n (X/H) = 0} e I ^f]{X/Ua : a e A and P n Ua = 0} e I ^ X/U{Ua : a e A Mid P n Ua = 0} e I ^ X/St(P, U) e I.
Therefore (X, t, I) is a strongly star /-compact space.
(2) ^ (1) Let (X, t, I) is a strongly star /-compact space and % = {Ha : a e A} be a family of
%=0
Then for the family U = {Ua = X/Ha : a e A} is a family of open sets and UU = X/(naeaHa) = X/0 = X.
Therefore, U is an open cover of X. But X is strongly star /-compact space. So, there exists a finite subset P C X such that X/St(P, U e I. ^ x/{j {U eU : P n U = 0} e I. ^ P{X/U : U e U mid P n U = 0} e I.
^ f]{H e % : P n (X/H) = 0} e I, which is a contradiction to the fact that the family % has n% = 0 □
6. Countably I-compact
Definition 5. An ideal space (X, t, I) is called an countably I-compact space if for every countable open coverU, there exists a finite subset {U\, U2, U3,..., } C U such th at X/(Uk=1Ui) e I
Theorem 7. Every closed subspace of a countably I-compact space is countably I-compact.
Proof. Let (A, ta) be a closed subspace of a countablv /-compact space (X,t,I). Let, Ua = {Uau : n e N} be a TA-open cover of A. Therefore, for every Ua e Ua there exist U e t such that Ua = U n A. Assume that U = {Un : Un n A e UA}
Clearly, V = UU (X/A) is an open cover of X. But X is countably /-compact. Therefore, it has finite subset {X/A, Vi, V2, V3,..., VK} C V such th at X/((Uki=lVi) U (X/A)) = = h e I
(Uk=1 Vi) U (X/A) U h = X A n {(Uk=lVi) U (X/A) U h} = A ^ (A n (Uk=1Vi)) U (A n (X/A)) U (A n h) = A ^ Uk=1(A n Vi)) U 0 U (A n h) = A ^Uk=iUAi U (A n Ii) = A ^ A/ Uk=i UAi = A n Ii e Ia
Therefore (A, ta)is an countable compact subspace of (X,t,I). □
Theorem 8. Every countably Ifin-compact space is strongly star Ifin-compact space.
PROOF. Suppose that (X,t,I) is countablv /-compact but not strongly star /jin-compact. Let U be an arbitrary open cover of X then for every finite subset B C X, X/St(B, U) e Ifin- i-e-X/St(B, u ) = 0'
Consider a point x0 e X
Set other points as xn e X/St({x0,x1, ...,xn-1},U) and construct a countable open cover V = {Vn = St(xn-i,U) : n e N}
Let, А = {xn_i :£ N}. If г/ £ A, we have an open set U £ U (since U is an open cover)such that у £ U. But у £ A, therefore, A nU = 0.
Let, xk-i £ A nU ^U Q St (xk-i, U) ^y £ St (xk-i, U) ^ у £ Vk for som e к £ N. Therefore, V is a countable cover of A. But A is a closed subspace of (X, т, I), A is also countablv //¿n-compact (by Theorem 7). Therefore, there exists finite subset {Vni, Vn2, Vn3,..., Vnp} Q V such that
AAULiVn,) = Ii £ Ifin
^ (u) и Ii = a
^A Q ^ v^Vni) и Ii.
But the construction of V, each V^ can contrnn only one element of A also I, is a finite subset of X A
Hence, (X, т, I) is ад strongly star ijin-compact space. □
Open Problem Does there exists a strongly star //¿n-compact space which is not countablv -f/m-compact.
7. Conclusion
compactness. This covering property can also be expressed in terms of family of closed sets by means of modified and idealized finite intersection property. All countablv //¿n-compact spaces are strongly star //¿n-compact space although their structures are very different. These topological properties can further be used in the study of selection principles and topological games involbing ideals.
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Получено: 28.02.2024 Принято в печать: 04.09.2024
