第 54 卷第 6 期
2019 年 12 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 54 No. 6
Dec. 2019
ISSN: 0258-2724
DOI：10.35741/issn.0258-2724.54.6.39
Regular article Mathematics
O
N
M
AXIMAL AND
M
INIMAL
S
ETS IN
T
OPOLOGICAL
S
PACES
a University of Al-Qadisiyah, College of Science, Department of Mathematics, Ad Diwaniyah,Raad Aziz Hussain Al-Abdulla Iraq E-mail*: [email protected]
Abstract
In this paper, we examine new classes of sets in topological spaces that are called maximal and minimal sets. Their relations are introduced and studied. Also, a recent class of functions has been found in topological spaces which use maximal and minimal sets. In addition, maximal and minimal sets are independent; the maximal and minimal sets cannot be equivalent under any conditions.
Keywords: maximal open sets, minimal open sets, maximal sets, minimal sets, topological spaces
摘要 在本文中，我们研究了拓扑空间中称为最大和最小集的新类集。 介绍并研究了它们之间的关系。 而且，在 使用最大和最小集的拓扑空间中发现了最近的一类功能。 另外，最大和最小集是独立的； 最大和最小集在 任何条件下都不能相等。 关键词: 最大开集，最小开集，最大集，最小集，拓扑空间
I.
I
NTRODUCTIONMaximal closed and minimal open sets were introduced in 2001, and 2003, respectively. These sets represent subclasses of open and closed of sets. In addition, maximal open and minimal closed sets were studied later in [3]-[5]. The purpose of this work is to introduce some definitions using maximal open and minimal closed sets which we call maximal sets and minimal sets. We consider some fundamental properties and establish some foundation for their theory. Also we introduce new functions which we call minimal closed continuous, minimal closed continuous and minimal irresolute functions and study the relations among them.
II. M
AXIMAL ANDM
INIMALS
ETS Definition 1.1: Let (X,τ) be a topologicalspace. Then
(1) A proper non-empty open subset U of X is considered to be a maximal open, (or simply an
M-open) set [2], if in case any open set to be contained in U is X or U.
(2) A proper non-empty closed-subset F of X is considered to be a minimal closed, (or simply an m-closed) set [1], in case any closed set to be contained in F is or F.
Definition 1.2: Let (X,τ) be a topological
space. Thus, a proper non-empty subset A of X is to be
1- a maximal set (or simply M-set) if is an M-open set.
2- a minimal set (or simply m-set) if is an m-closed set.
Theorem 1.3 [2] Let (X,τ) be a topological
space, and U be a subset of X. Therefore, U is an m-closed if and only if is an M-open set.
Theorem 1.4: Let (X,τ) be a topological
1- A is an set and open, if and only if A is an M-open set.
2- A is an m-set, and closed if and only if, A is an m-closed set.
Proof:
1- Let A be an M-set and open. Then is an M-open set and A= . Therefore, A is an open set. In the opposite case, let A be an M-open set. Then A is M-open and hence is an M-open set. Therefore, A is an M-set.
2- Let A be an m-set and closed. Then, is an m-closed set and A= . Therefore A is an m-closed set. Oppositely, let A be an m-closed set. Then, A is to be a closed set and so is an m-closed set. Therefore, A is an m-set.
Example 1.5: Let X= and τ=
{∅, 𝑋, {𝑎, 𝑏}, {𝑎}} be a topology on X. Let A= . Then = . Thus, A is an M-set. Since A is not an open set, then A is not M-open. Let B= . Then . Thus B is an set. Since B is not a closed set, then B is not m-closed.
Theorem 1.6: Let (X,τ) be a topological
space and U be a subset of X. Then U is an m-set if and only if is an M-set.
Proof: Let U be an m-set and W be an open
set such that W. Then . Since U is an m-set, then is an m-closed. Therefore, . Hence W=X or W= .
Therefore is an M-open set. Then is an M-set. Conversely, let be an M-set and F be a closed set such that F . Then
. Since is an M-set, then is an M-open. Therefore, . Hence F= or F= . Therefore is an m-closed set. Then U is an m-set.
Example 1.7: Let X= and τ=
be a topology on X. If A= . Then A is an M-set and an M-open set, but it is not an m-set
and not m-closed. If B= . Then B is an m-set and m-closed set, but it is not an M-set and not an M-open set.
Theorem 1.8: Let (X,τ) be a topological
space and A X. Then
1- If A is an M-set and B is a subset of X such that B X and A B, then .
2- If A is an m-set and B be a non-empty subset of X, B A then .
Proof:
1- Let A be an M-set, B be a subset of X such that B X and A B. Then is an M-open set and . Since B X, then
and hence .
2- Let A be an m-set, B be a non-empty subset of X such that B A. Then
is an m-closed set and . Since B , then and hence
Theorem 1.9: Let (X,τ) be a topological
space. If A is an m-set and F is a closed set such that A , then .
Proof: Since A is an m-set, then is an m-closed set. Also since is a closed set and
, then . Hence .
Corollary 1.10: Let (X,τ) be a topological
space. If A is an m-set and W is an open set such that , then .
Proof: The proof follows from theorems 1.6
and 1.9.
Corollary 1.11: If F and K are m-sets in a
topological space (X,τ) such that then .
Proof: Let F and K be m-sets such that
.Then is an m-closed sets such that . Therefore, by theorem 1.9, . Hence . By the same way we can prove that . Then
Corollary 1.12: If U and V are M-sets in a
topological space (X,τ) such that
Proof: Let U and V be m-sets such that
that U∪ 𝑊𝑜 ≠ 𝑋. Therefore by corollary 1.10, . Hence . By the same way we can prove that . Then
Theorem 1.13: Let (X,τ) be a topological
space and A be an m-set. If , then for every F closed set which contains x.
Proof: Let F be a closed-set such that . Since A is an m-set, then is an m-closed set. Then is a closed set such that and . Since and is an m-closed set, then
. Therefore, .
Corollary 1.14: Let (X,τ) be a topological
space and A be an m-set. If , then for every W open set such that .
Proof: Let A be an m-set such that . Then by theorem 1.6 is an m-set such that x∈ 𝐴𝑐. Hence by theorem 1.13 for every F closed set which contains x. Hence
such that where .
Corollary 1.15: Let (X,τ) be a topological
space, A be an m-set and . Then is a closed set and .
Proof: By theorem 1.13, we obtain
for every closed set F which contains x. Hence .
Thus ⊆ ∩ { 𝐹: 𝐹 is a closed set and . Since is a closed set and , then is a closed set and .
Theorem 1.16: Let (X,τ) be a topological
space, A be an m-set and . Then is an open set and .
Proof: By corollary 1.14, we get for every W open set such that . Hence
.Thus is an open set and . Since is an open set and , then. is an open set and .
Theorem 1.17: Let (X,τ) be a topological
space, A be an M-set and . Then for any open set W such that .
Proof: Let . Then for any open set W such that . Then by corollary 1.10, A =X. Therefore
Theorem 1.18: Let (X,τ) be a topological
space, A be an m-set and x A. Then A F= for any closed set F such that x F.
Proof: Let x A, then A F for every closed set F such that x F. Then by theorem 1.9, A F= .
Theorem 1.19: Let (X,τ) be a topological
space and A be an m-set. Then either .
Proof: Let A be an m-set. Suppose . Then there exists x X such that x . Thus x
A. Then by theorem 1.17, for any open set W such that x W. Since is an open set and x∈ , then . Thus
. Therefore
Corollary 1.20: Let (X,τ) be a topological
space and A be an m-set. Then either
Proof: The proof follows from theorems 1.6
and 1.19.
Corollary 1.21: Let (X,τ) be a topological
space and A be an m-set and M-set. Then A is an open and closed set.
Proof: Since is an m-closed set and is an M-open set, then .
Therefore by theorem 1.19 and corollary 1.20, A is open and closed.
Corollary 1.22: If a topological space (X,τ)
has a set G which is both m-set and M-set, then X is disconnected.
Proof: The proof follows from corollary 1.21. Theorem 1.23: Let (X,τ) be a topological
space. If X contains an M-set A and an open set B such that B A, then .
Proof: Let A be an m-set, then is an M-open set. Since and
Corollary 1.24: Let (X,τ) be a topological
space. If X contains an m-set A and a closed set B such that A , then .
Proof: The proof follows from theorems 1.6
and 1.23.
Corollary 1.25: Let (X,τ) be a topological
space. If X contains M-set A and m-set B such that , then X is disconnected.
Proof: Since , then and
. Since A is an m-set and is an open set such that , then by theorem 1.23, =X. Also since B is an m-set and is a closed set such that B , then by corollary 1.24, . Therefore A and B form a separation for X. Then X is disconnected set.
Theorem 1.26: Let (X,τ) be a topological
space and A be an m-set. If S is a non-empty subset of A, then .
Proof: Let ,
then
contradiction. In case A ∩ 𝑆̅ ≠ ∅ , then by
theorem 1.9
.
Corollary 1.27: Let (X,τ) be a topological
space and A be an m-set. If S is a non-empty set disjoint from A, then .
Proof: The proof follows from theorems 1.26
and 1.6.
Theorem 1.28: Let (X,τ) be a topological
space, A be an m-set and B be a subset of X such that A⊂ 𝐵. Then 𝐵 ̅ =X.
Proof: Since A B, then there exists a non-empty subset S of such that B=A S, then by corollary 1.27 = . Therefore,
Corollary 1.29: Let (X,τ) be a topological
space, A be an m-set and B be a subset of X such that . Then
Proof: The proof follows from theorems 1.6
and 1.28
Theorem 1.30: Let (X,τ) be a topological
space and A,B X. If A is an m-set and , then B is an m-set.
Proof: Let A be an m-set and B A. Then is an m-closed set. Let F be a closed set such that . Then and hence . If F= , then . Therefore, B is an m-set.
Corollary 1.31: If A is a non-empty subset of
a topological space (X,τ), then A is an m-set if and only if is an m-set.
Proof: the proof follows from the definition
of an m-set and theorem 1.30.
Theorem 1.32: Let (X,τ) be a topological
space and A,B X. If A is an m-set and A B X, then B is an m-set.
Proof: Let A be an m-set and A B such that B X. Then is an M-open set. Let G be an open set such that . Then and hence G=X or . If , then
. Therefore B is an m-set.
Corollary 1.33: Let A be a subset of a
topological space (X,τ) such that . Then A is an m-set if and only if is an m-set.
Proof: the proof follows from theorem 1.32
and the definition of an M-set.
Corollary 1.34: Let (X,τ) be a topological
space, A and B be subsets of X. If A is an m-set such that , then B is an m-set.
Proof: The proof follows from corollary 1.31
and theorem 1.30.
Corollary 1.35: Let (X,τ) be a topological
space, A and B be subsets of X. If A is an m-set such that is an m-set.
Proof: The proof follows from corollary 1.33
and theorem 1.32.
Corollary 1.36: Let (X,τ) be a topological
space, A and B be subsets of X. If A is an m-set such that A B , then A B is an m-set.
Proof: The proof follows from corollary 1.30. Corollary 1.37: Let (X,τ) be a topological
space, A and B be subsets of X. If A is an m-set such that A , then A B is an m-set.
Proof: The proof follows from theorem 1.32. Theorem 1.38: Let (X,τ) be a topological
m-set, then = for any set B such that A B .
Proof: Let A be an m-set. Then by corollary
1.31, is an m-closed set. Hence . Since is an
m-closed set,
, then .
Theorem 1.39: Let (X,τ) be a topological
space, A,B ⊆ X and 1et A be an m-set. Then for any set B such that A
Proof: Let A be an m-set .Then is an M-open set. Since is an M-open set,
and A B
Theorem 1.40: Let A be a non-empty proper
subset of a topological space (X,τ). Then the following statements are equivalent.
1- If A is open, then A is an m-set. 2- If A is closed, then A is an m-set.
Proof: The proof follows from theorem 1.6. Theorem 1.41: Let A be a non-empty proper
subset of a topological space (X,τ). Then the following statements are equivalent:
1- If A is M-open, then A is an m-set. 2- If A is m-closed, then A is an m-set.
Proof: The proof follows from theorems 1.3
and 1.6.
Theorem 1.42: Let A be a subset of a
topological space (X,τ). Then the following statements are equivalent:
1- If A is an m-set, then A is M-open. 2- If A is an m-set, then A is m-closed.
Proof: The proof follows from theorems 1.3
and 1.6
Theorem 1.43: Let A be a subset of a
topological space (X,τ). Then the following statements are equivalent:
If A is an m-set, then A is an m-set.
Proof: The proof follows from theorem 1.6. Theorem 1.44: Let (Y,τ ) be a closed subspace of a topological space (X,τ).
If A is an m-set in X such that Y⊈ 𝐴 ̅̅̅and
, then .
Proof: Let F be a closed set in Y such that F
. Then F is closed in X and F = . Since A is an
m-set in X, then . In case F , then
. Therefore, . Then is an m-set in Y.
Theorem 1.45: Let (Y,τ ) be an open and a
closed subspace of a topological space (X,τ). If A is an m-set in X such that Y and
, then A Y is an m-set in Y.
Proof: Let G be an open set in Y such that
. Then G is open in X and
. Hence Since A is an m-set, then
either In case
, we obtain G=Y. In case , we get
. Therefore, . Then is an m-set in Y.
Theorem 1.46: Let (Y,τ ) be an open and a
closed subspace of a topological space (X,τ). If every non-empty proper open subset of X is an m-set in X, then every non-empty proper open subset of Y is an m-set in Y.
Proof: Let U be a non-empty proper open
subset of Y, then U is a non-empty proper open subset of X. Thus U is an m-set in X. Since , then by theorem 1.45, U is an m-set in Y.
Theorem 1.47: Let (Y,τ ) be a closed subspace of a topological space (X,τ). If every non-empty proper closed subset of X is an m-set of X, then every non-empty proper closed subset of Y is an m-set in Y.
Proof: Let F be a non-empty closed subset of
Y. Then F is a non-empty proper closed subset of X. Thus F is an m-set in X. Since F
, then by theorem 1.44, F is an m-set in Y.
III. (M
INIMALC
LOSED-C
ONTINUOUS,
C
LOSEDM
INIMAL-C
ONTINUOUS ANDM
INIMALI
RRESOLUTE)
F
UNCTIONSDefinition 2.1: Let (X,τ) and (Y,τ`) be
topological spaces. A function 1- minimal closed-continuous (briefly mc-continuous) if (A) is an m-closed in X for every m-set A in Y.
2- closed minimal-continuous (briefly cm-continuous) if (A) is an m-set in X for every m-closed set A in Y.
3- minimal irresolute (briefly m-irresolute) if (A) is an m-set in X for every m-set A in Y.
The proof of theorems 2.2, 2.3 and 2.4 follows from theorems 1.3 and 1.6.
Theorem 2.2: Let (X,τ) , (Y,τ`) be topological
spaces and be a function. Then the following statements are equivalent:
1- f is mc-continuous
2- (A) is M-open in X for every M-set A in X.
Theorem 2.3: Let (X,τ) and (Y,τ`) be
topological spaces and f:X Y be a function. Then the following statements are equivalent:
1- f is cm-continuous
2- (A) is an m-set in X for every M-open set A in Y.
Theorem 2.4: Let (X,τ) , (X,τ`) be topological
spaces and be a function. Then the following statements are equivalent:
1- f is m-irresolute.
2- (A) is an m-set in X for every M-set A in Y.
Theorem 2.5: Let (X,τ) and (Y,τ`) be
topological spaces and be a function. Then:
1- If f is mc-continuous, then it is m-irresolute.
2- If f is m-irresolute, then it is cm-continuous.
Thus, if f is mc-continuous, then it is cm-continuous.
Proof: the proof holds since every m-closed
set is an m-set.
Example 2.6: Let X with τ and Y be with τ`= {∅, {𝑎}, {𝑏}, {𝑎, 𝑏}, 𝑌}. Then is an
m-set in X, but it is not m-closed. Let f:X Y be an identity function. Then f is m-irresolute, but it is not mc-continuous, since by theorem 2.2, for the M-set in Y,
which is not M-open in X.
Theorem 2.7: Let (X,τ) and (Y,τ`) be
topological spaces , f:X Y be a function and X has the property that every m-set is m-closed. If f is m-irresolute, then it is mc-continuous.
Proof: the proof follows from the hypothesis. Example 2.8: Let X be with τ={ ∅, 𝑋, {𝑎}} 𝑎𝑛𝑑 𝑌 = {𝑎, 𝑏, 𝑐} be with τ`= {∅, 𝑌, {𝑎}}. Then is an m-set in Y but it is not m-closed. Let f:X Y be a function such that f(a)=a and f(b)=b. Then f is cm-continuous, but it is not m-irresolute, since by theorem 2.4, for the M-set in Y,
.
Theorem 2.9: Let (X,τ) and (Y,τ`) be
topological spaces, f:X Y be a function and Y has the property that every m-set is m-closed. If f is cm-continuous, then it is m-irresolute.
Proof: The proof follows from the hypothesis. Example 2.10: In example 2.8, f is
ccontinuous. Since in this example f is not m-irresolute and by theorem 2.5 every mc-continuous function is m-irresolute, then f is not mc-continuous.
Theorem 2.11: Let (X,τ) and (Y,τ`) be
topological spaces, f:X Y be a function and X,Y have the property that every set is m-closed. If f is cm-continuous, then it is mc-continuous.
Proof: the proof follows from the hypothesis. Theorem 2.12: Let (X,τ) and (Y,τ`) be
topological spaces and f:X Y be bijective, closed and continuous function. If A is an m-set in X, then f(A) is an m-set in Y.
Proof: Let A be an m-set in X. Then is an m-closed set in X. Let F be a non-empty closed set in Y such that F . Since f is a closed function, then . Since f is injective function, then . But f is a continuous function and is an m-closed set, then either or
. If and since f is a surjective function, then F . If
and since f is a surjective, continuous and closed function, then F . Therefore, is an m-closed set. Hence f(A) is an m-set.
Corollary 2.13: Let (X,τ) and (Y,τ`) be
topological spaces and f:X Y be a bijective, closed and continuous function. If A is an m-set in X, then f(A) is an M-set in Y.
Proof: Let A be an m-set in X. Then X-A is
an m-set in X. Therefore, by theorem 2.12, f(X-A) is an m-set in Y. Hence Y-f(X-f(X-A) is an M-set in Y. Since f is a bijective function, then f(A) is an m-set in Y.
Theorem 2.14: Let (X,τ) and (Y,τ`) be
topological spaces and f:X Y be an injective open and continuous function. If A is an m-set in Y, then is an m-set in X.
Proof: Let A be an m-set in Y. Then is an m-closed set in Y. Let f be a non-empty closed set in X such that F . Since f is continuous function, then
. Since f is closed function and is an m-closed set in Y, then either f(F) or . If f (F) then F . If f(f) and since f is an injective open and continuous function, then . Therefore, is an m-set in A.
Corollary 2.15: Let (X,τ) and (Y,τ`) be
topological spaces and f:X Y be an injective, open and continuous function. If A is an m-set in Y, then is an m-set in X.
Proof: Let A be an m-set in Y. Then Y-A is
an m-set in Y. Therefore, by theorem 2.14, is an m-set in X. Hence is an m-set in X. Therefore,
.
Theorem 2.16: Let (X,τ), (Y,τ`) and (Z,τ )
be topological spaces. Then:
1- If f:X Y is mc-continuous and g:Y Z is m-irresolute, then gof is
mc-continuous and hence gof is cm-mc-continuous and m-irresolute.
2- If f:X Y is m-irresolute and g:Y Z is mc-continuous, then gof is m-irresolute and hence gof is cm-continuous.
3- If f:X Y is m-irresolute and g:Y Z is cm-continuous, the gof is cm-continuous. 4- If f:X Y is mc-continuous and g:Y Z is continuous, then gof is cm-continuous.
5- If f:X Y is cm-continuous and g:Y Z is mc-continuous, then gof is m-irresolute and hence gof is cm-continuous.
Proof: The proof follows from theorem 2.5
since every m-closed set in the m-set.
Example 2.17: Let X=Z and Y be with τ , μ
and ϻ .
Let f:X Y such that f(a) a, f (b) b, f(c) c f(d) c and g:Y Z such that g(a) a, g(b) b, g(c) c. Then f is m-irresolute, continuous, mc-continuous and g is cm-continuous, but gof is not m-irresolute, since is an m-set in Z but is not an m-set in X. Also by theorem 2.5, gof is not mc-continuous.
Example 2.18: Let X=Y=
τ , μ and ϻ . Let f:X Y be the identity map and g:Y Z be such that g(a) =a, g(b)= b and g(c)= c. Then f is m-irresolute, continuous, mc-continuous and g is cm-continuous, but gof is not m-irresolute, since is an m-set in Z, but is not an m-set in X. Also by theorem 2.5, gof is not mc-continuous.
Example 2.19: Let X=Y=Z= be
with τ , μ
and ϻ . Let f:X Y and g:Y Z be the identity functions. Then f is m-irresolute, cm-continuous and g is m-m-irresolute, cm-continuous, continuous, but gof is not
mc-continuous, since is an m-set in Z but is not m-closed in X. Example 2.20: Let X= be with τ , μ= and ϻ so that
f(a)=a, f(b)=b, f(c)=c and g:Y Z be the identity functions. Then f is cm-continuous, g is m-irresolute and cm-continuous, but gof is not cm-continuous, since is m-closed in Z, but is not an m-set in X. Also by theorem 2.5, gof is not m-irresolute and not mc-continuous.
In examples 2.17, 2.18, 2.19 and 2.20 we show that theorems 2.21, 2.22, 2.23, 2.24 and 2.25 are not true in general.
Theorem 2.21: Let (X,τ), (Y,τ`) and (Z,τ``)
be topological spaces such that X has the property that every m-set is m-closed. Then:
1- If f:X Y is m-irresolute and g:Y Z is mc-continuous, then gof is mc-continuous. 2- If f:X Y is cm-continuous and g:Y
Z is continuous, then gof is mc-continuous.
Proof: The proof follows from the hypothesis
since every m-closed set is an m-set.
Theorem 2.22: Let (X,τ), (Y,τ`) and (Z,τ``)
be topological spaces such that Y has the property every m-set is m-closed. If f:X Y is cm-continuous and g:Y is m-irresolute, then gof is m-irresolute and hence gof is cm-continuous.
Proof: the proof follows from the hypothesis
and theorem 2.5.
Theorem 2.23: Let (X,τ) , (Y,τ`) and (Z,τ``)
be topological spaces such that Z has the property that every m-set is m-closed. Then:
1- If f:X Y is m-irresolute and g:Y Z is ccontinuous, then gof is m-irresolute.
2- If f:X Y is mc-continuous and g:Y Z is cm-continuous, then gof is mc-continuous and hence gof is m-irresolute.
Proof: The proof follows from the hypothesis
and theorem 2.5.
Theorem 2.24: Let (X,τ), (Y,τ`) and (Z,τ``)
be topological spaces so that X and Y have the property that every m-set is m-closed. If f: X
Y is cm-continuous and g:Y Z is m-irresolute, then gof is mc-continuous.
Proof: The proof follows from the hypothesis. Theorem 2.25: Let (X,τ), (Y,τ`) and (Z,τ``)
be topological spaces so that X and Z have the property that every m-set is m-closed. If f:X Y is m-irresolute and g:Y Z is cm-continuous, then gof is mc-continuous.
Proof: the proof follows from the hypothesis.
IV. C
ONCLUSIONSIn the present paper, we have defined new kinds of sets which are called minimal sets and maximal sets. A maximal open-set is a maximal set but not conversely and a minimal-closed set is a minimal set, but not conversely, too. Also we defined new kinds of continuous functions by using these new kinds of sets. Properties and characterizations of the new kinds of sets and functions are investigated.
