Download On Semi*- Closed Sets

Asian Journal of Current Engineering and Maths1: 4 Jul – Aug (2012) 173 – 176.
Contents lists available at www.innovativejournal.in
ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage : http://www.innovativejournal.in/index.php/ajcem
On Semi*- Closed Sets
A. Robert1, S. Pious Missier2
1Department
2P.G.
of Mathematics, Aditanar College, Tiruchendur, India-628216.
Department of Mathematics, V.O.Chidambaram College, Thoothukudi, India-628008.
ARTICLE INFO
ABSTRACT
Corresponding Author
A. Robert
Department of Mathematics,
Aditanar College, Tiruchendur,
India-628216.
Abstract: In this paper we introduce a new class of sets, namely semi*-closed
sets. We give characterizations for semi*-closed sets. We define the semi*closure, semi*-limit points and semi*-derived set of a subset. Further we
investigate fundamental properties of semi*-closed sets.
Key Words: Semi-Closed, SemiClosure, Semi-Limit Point, Semi*Open, Semi*-Closed, Semi*-Closure,
Semi*-Limit Point.
Mathematics Subject Classification: 54A05.
©2012, AJCEM, All Right Reserved.
1. INTRODUCTION
N.Levine [7] introduced semi-open sets in topological
spaces in 1963. N.Biswas [1] defined semi-closed sets.
S.G.Crossley and S.K.Hildebrand [2] defined semi-closure of
a set. N.Levine [8] defined and studied generalized closed
sets in 1970. P.Das [3] defined semi-limit point of a subset.
The semi-derived set of a subset of a topological space was
also defined and studied by him. W.Dunham [5] introduced
the concept of generalized closure and defined a new
topology τ* and investigated some of their properties. G.Di
Maio and T.Noiri [4] defined and studied semi-regular
subsets. Quite recently the authors [11] introduced and
studied semi*-open sets using the generalized closure
operator.
In this paper, we introduce the concept of semi*closed sets. We further show that the class of semi*-closed
sets lies between the class of semi-closed sets and the class
of closed sets. We give characterizations of semi*-closed
sets. We investigate many fundamental properties of
semi*-closed sets. We also define the semi*-closure of a
subset and study their properties.
2. PRELIMINARIES
Throughout this paper (X, τ) will always denote a
topological space on which no separation axioms are
assumed, unless explicitly stated. If A is a subset of the
space (X, τ), Cl(A) and Int(A) denote the closure and the
interior of A respectively. Also 𝓕 denotes the class of all
closed sets in the space (X, τ).
Definition 2.1: A subset A of a space X is
(i) generalized closed (briefly g-closed) [8] if Cl(A)⊆U
whenever A⊆U and U is open in X.
(ii) generalized open (briefly g-open) [8] if X\A is g-closed
in X.
Definition 2.2: If A is a subset of X,
(i) The generalized closure [5] of A is defined as the
intersection of all g-closed sets in X containing A and is
denoted by Cl*(A).
(ii) the generalized interior of A is defined as the union of
all g-open sets in X that are contained in A and is denoted
by Int*(A).
Definition 2.3: A subset A of a topological space (X, τ) is
(i) semi-open [7] (resp. semi*-open [11] ) if there is an
open set U in X such that U⊆A⊆Cl(U) (resp. U⊆A⊆Cl*(U))
or equivalently if A⊆Cl(Int(A)) (resp. A⊆Cl*(Int(A))).
(ii) semi-closed [1](resp. pre-closed [9] and -closed
[10]) if Int(Cl(A))⊆A ( resp. Cl(Int(A)⊆A and
Cl(Int(Cl(A)))⊆A ).
The class of all semi-open (resp. semi-closed, semi*-open)
sets is denoted by SO(X, τ) (resp. SC(X, τ), S*O(X, τ)).
Definition 2.4: [8] A topological space X is T1/2 if every gclosed set in X is closed.
Definition 2.5: [5] If (X, τ) is a topological space, let τ* be
the topology on X defined by the Kuratowski closure
operator Cl*. That is, τ*={U⊆X : Cl*(X\U)=X\U}.
Theorem 2.6: [5] If (X, τ) is a topological space, then (X, τ*)
is T1/2.
Definition 2.7: A space X is locally indiscrete [12] if every
open set in X is closed.
Definition 2.8: [6] The topology on the set of integers
generated by the set S of all triplets of the form {2n-1, 2n,
2n+1} as sub base is called the Khalimsky topology and it
is denoted by κ. The set S∪{2n+1:n∈ℤ} is a base for the
topology κ. The digital line equipped with the Khalimsky
topology is called the Khalimsky line. A subset A is open in
(ℤ, κ) iff whenever x in A is even, x-1, x+1∈A. A subset A is
closed in (ℤ, κ) iff whenever x in A is odd, x-1, x+1∈A. The
topological product of two Khalimsky lines (ℤ, κ) is called
the Khalimsky plane (ℤ2, κ2).
Theorem 2.9: [11] every open set is semi*-open.
Theorem 2.10:[11] Every semi*-open set is semi-open.
3. SEMI*-CLOSED SETS
Definition 3.1: A subset A of a space (X, τ) is called a
semi*-closed set if X\A is semi*-open.
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Notation: The class of all semi*-closed sets in (X, τ) is
denoted by S*C(X, τ) or simply S*C(X).
Definition 3.2: The semi*-closure of A is defined as the
intersection of all semi*-closed sets in X containing A. It is
denoted by s*Cl(A).
Definition 3.3: Let A⊆X. An element x∈X is called a semi*adherent point of A if every semi*-open set in X containing
x intersects A.
Definition 3.4: Let A⊆X. An element x∈X is called a semi*limit point of A if every semi*-open set in X containing x
intersects A in a point different from x.
Definition 3.5: The set of all semi*-limit points of A is
called the semi*-Derived set of A.
It is denoted by Ds*[A].
Definition 3.6: A subset A of X is called semi*-regular if it
is both semi*-open and semi*-closed.
Theorem 3.7: A subset A of a space (X, τ) is semi*-closed if
and only if there is a closed set F in (X, τ) such that
Int*(F)⊆A⊆F.
Proof: Necessity. Suppose A is semi*-closed. Then X\A is
semi*-open. Then there exists an open set U in X such that
U⊆X\A⊆Cl*(U) which implies X\U⊇A⊇X\Cl*(U). Note that
in
any
space,
X\Cl*(U)=Int*(X\U).
Therefore
X\U⊇A⊇Int*(X\U) where X\U is closed in X. Sufficiency.
Suppose there is a closed set F in (X, τ) such that
Int*(F)⊆A⊆F which implies X\Int*(F)⊇X\A⊇ X\F. Since
X\Int*(F)=Cl*(X\F), we have Cl*(X\F)⊇X\A⊇X\F where X\F
is an open set. Hence by Definition 2.3, X\A is semi*-open.
Therefore A is semi*-closed.
Remark 3.8: In any space (X, τ), ϕ and X are semi*-closed
sets.
Theorem 3.9: A⊆X is semi*-closed if and only if
Int*(Cl(A))⊆A.
Proof: Note that if A⊆B then X\A⊇X\B and
Cl*(X\A)=X\Int*(A) and X\Cl*(A)=Int*(X\A).
A is semi*-closed if and only if X\A is semi*-open. That is, if
and only if X\A⊆Cl*(Int(X\A)) (By Definition2.3(i)). That
is, if and only if Int*(Cl(A))⊆A.
Theorem 3.10: If A is semi*-closed in X and B⊆X is such
that Int*(A)⊆B⊆A. Then B is semi*-closed in X.
Proof: Since A is semi*-closed, X\A is semi*-open. Now
Int*(A)⊆B⊆A which implies
X\Int*(A)⊇X\B⊇X\A. That is, Cl*(X\A)⊇X\B⊇X\A.
Therefore by Theorem 3.39 [11], X\B is semi*-open. Hence
B is semi*-closed.
Theorem 3.11: If {Aα} is a collection of semi*-closed sets,
then ⋂Aα is also a semi*-closed set.
Proof: Since each Aα is semi*-closed, X\Aα is semi*-open.
By Theorem 3.11[11], ∪(X\Aα) is semi*-open. That is,
X\(⋂𝐴𝛼 ) is semi*-open. Hence ⋂Aα is semi*-closed.
Corollary 3.12: If A is semi*-closed and U is semi*-open in
X, then A\U is semi*-closed in X.
Proof: Follows from A\U=A⋂(X\U).
Remark 3.13: Union of two semi*-closed sets need not be
semi*-closed as seen from the following example.
Example 3.14: Let X={a, b, c, d} with τ={ϕ, {a}, {b}, {a, b},
{a, b, c}, X}. In the space (X, τ), the subsets A={a, c} and B={b
,c} are semi*-closed but A∪B={a, b, c}is not semi*-closed.
Theorem 3.15: If A is semi*-closed in X and B is closed in
X, then A∪B is semi*-closed in X.
Proof: Since A is semi*-closed, X\A is semi*-open in X. Also
X\B is open. By Theorem 3.10[11], (X\A)⋂(X\B)=X\(A∪B)
is semi*-open in X. Hence A∪B is semi*-closed in X.
Theorem 3.16: If A is any subset of a space X, then s*Cl(A)
is semi*-closed. In fact s*Cl(A) is the smallest semi*-closed
set in X containing A.
Proof: Follows from Definition 3.1 and Theorem 3.11.
Theorem 3.17: A is semi*-closed if and only if s*Cl(A)=A.
Proof: A is semi*-closed implies s*Cl(A)=A is obvious. On
the other hand, let s*Cl(A)=A.
By Theorem 3.16, s*Cl(A) is semi*-closed and hence A is
semi*-closed.
Theorem 3.18: Every closed set is semi*-closed.
Proof: Let A be closed. Then X\A is open. By Theorem 2.9,
X\A is semi*-open. Hence A is semi*-closed.
Remark 3.19: The converse of Theorem 3.18 is not true
as shown by the following example.
Example 3.20: In the space (X, τ) of Example 3.14, the
subsets A={a, c} and B={b, c} are semi*-closed but not
closed.
Corollary 3.21: If A is semi*-closed and F is closed in X,
then A⋂F is semi*-closed in X.
Proof: Since F is closed, by Theorem 3.18, F is semi*closed. Then by Theorem 3.11, A⋂F is semi*-closed.
Corollary 3.22: If A is semi*-closed and U is open in X, then
A\U is semi*-closed in X.
Proof: Follows from A\U=A⋂(X\U) and from Corollary
3.21.
Theorem 3.23: Every semi*-closed set is semi-closed.
Proof: If A is semi*-closed, X\A is semi*-open. By
Theorem 2.10, X\A is semi-open. Hence A is semi-closed.
Remark 3.24: The converse of Theorem 3.23 is not true
as shown by the following example.
Example3.25: Consider the space in Example 3.14. In the
space (X, τ), the subsets {a, d} and
{b, d} are semi-closed but not semi*-closed.
Theorem 3.26: In any topological space (X, τ), 𝓕⊆S*C(X,
τ)⊆SC(X, τ).
Proof: Follows from Theorem 3.18 and Theorem 3.23.
Remark 3.27: (i) If (X, τ) is a locally indiscrete space, then
𝓕=S*C(X, τ)=SC(X, τ).
(ii) In the Sierpinski space (X, τ), where X={0, 1} and τ = {,
{1}, X}, 𝓕=S*C(X, τ)=SC(X, τ). (iii) The inclusions in
Theorem 3.26 may be strict and equality may also hold.
This can be seen from the following examples.
Example 3.28: Consider the space (X, τ) where X={a, b, c,
d} and τ ={ϕ, {a} ,{ b, c, d}, X}
𝓕={ϕ, {a}, {b, c, d}, X}. Here 𝓕=S*C(X, τ)=SC(X, τ).
Example 3.29: Consider the space (X, τ) where X={a, b, c,
d} and τ ={ϕ, {a}, {b}, {a, b},
{b, c}, {a, b, c}, X}. Here 𝓕={ϕ, {d}, {a, d}, {c, d}, {a, c, d}, {b, c,
d}, X}.
SC(X, τ)=S*C(X, τ)={ϕ,{a}, {c}, {d}, {a, c}, {a, d}, {b, c}, {c,
d},{a, c, d}, {b, c, d}, X}.
Here 𝓕 ⊊ S*C(X, τ)=SC(X, τ).
Example 3.30: Consider the space (X, τ) where X={a, b, c,
d} and τ={ϕ, {a, b}, X}.
𝓕={ϕ, {c, d}, X}; SC(X, τ) ={ϕ, {c}, {d}, {c, d}, X}; S*C(X,
τ)={ϕ, {c, d}, X}.
Here 𝓕=S*C(X, τ)⊊SC(X, τ).
Example 3.31: Consider the space (X, τ) where X={a, b ,c,
d} and τ ={ϕ, {a} ,{a, b, c}, X}
𝓕={ϕ, {d}, {b, c, d}, X}; SC(X, τ)={ϕ, {b}, {c}, {d}, {b, c}, {b, d},
{c, d}, {b, c, d}, X};
S*C(X, τ)={ϕ,{d}, {b, c}, {b, c, d}, X}. Here 𝓕⊊S*C(X, τ)⊊SC(X,
τ).
Remark 3.32: In a T1/2 space, the class of semi*-closed sets
and the class of semi-closed sets coincide. In particular, in
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Robert et.al/On Semi*- Closed Sets
the Khalimsky line and in the real line with usual topology
the semi*-closed sets and the semi-closed sets coincide. In
any topological space, X\{x} is semi*-closed if and only if it
is closed.
Remark 3.33: The concepts of semi*-closed sets and αclosed sets are independent as seen from the following
examples:
Example 3.34: In the space (X, τ) where X={a, b, c, d} and
τ={ϕ, {a}, {a, b, c}, X}, the subsets {b}, {c}, {b, d} and {c, d}
are α-closed but not semi*-closed.
Example 3.35: In the space (X, τ) where X={a, b, c, d} and
τ={ϕ,{a},{b},{a, b}, {a, b, c}, X}, the subsets {a, c} and {b, c}
are semi*-closed but not α-closed.
Remark 3.36: The concepts of semi*-closed sets and preclosed sets are independent as seen from the following
examples:
Example 3.37: In the space (X, τ) where X={a, b, c, d} and
τ={ϕ, {a}, X}, the subsets {b, d} and {c, d} are pre-closed but
not semi*-closed.
Example 3.38: In the space (X, τ) where X ={a, b, c, d}and
τ={ϕ, {a}, {b}, {a, d}, {b, d},
{a, b, c}, X}, the subsets {a, c} and {b, c} are semi*-closed but
not pre-closed.
From the above discussions we have the following diagram:
Diagram 3.39:
Theorem 3.40: In any topological space (X, τ) ,the
following results hold:
(i) s*Cl(ϕ)=ϕ.
(ii) s*Cl(X)=X.
If A and B are subsets of X,
(iii) A⊆s*Cl(A).
(iv) A⊆B⟹s*Cl(A)⊆s*Cl(B).
(v) s*Cl(s*Cl(A))=s*Cl(A). That is, the operator s*Cl is
idempotent.
(vi) A⊆sCl(A)⊆s*Cl(A)⊆Cl(A).
(vii) s*Cl(A∪B)⊇s*Cl(A)∪s*Cl(B).
(viii) s*Cl(A⋂B)⊆s*Cl(A)⋂s*Cl(B).
(ix) Cl(s*Cl(A))=Cl(A).
(x) s*Cl(Cl(A))=Cl(A).
Proof: (i), (ii), (iii) and (iv) follow from Definition 3.2. (v)
follows from Theorem 3.16 and Theorem 3.17. (vi) follows
from Theorem 3.18 and Theorem 3.23. (vii) and (viii)
follow from (iv) above. From (iii) above, we have A⊆s*Cl(A)
and hence Cl(A)⊆Cl(s*Cl(A)). Also from (vi) above, we have
Cl(A)⊇s*Cl (A) and hence Cl (A)⊇Cl(s*Cl(A)).Therefore
Cl(s*Cl(A))=Cl(A). This proves (ix). Clearly, (x) follows from
the fact that Cl (A) is closed and hence semi*-closed and by
invoking Theorem 3.17.
Remark 3.41: In (vi) of Theorem 3.40, the inclusions may
be strict and equality may also hold. This can be seen from
the following example.
Example 3.42: Let X={a, b, c, d, e, f, g} be endowed with
τ={ϕ, {a}, {b}, {a, b}, {a, b, e},
{a, c, d}, {b, f, g}, {a, b, c, d}, {a, b, f, g}, {a, b, c, d, e}, {a, b, e, f,
g}, {a, b, c, d, f, g}, X}.Here 𝓕={ϕ, {e}, {c, d}, {f, g}, {c, d, e}, {e,
f, g}, {a, c, d, e}, {b, e, f, g}, {c, d, f, g}, {c, d, e, f, g}, {a, c, d, e, f,
g}, {b, c, d, e, f, g}, X}; SC(X, τ)={ϕ, {c}, {d}, {e}, {f}, {g},{c, d},
{c, e}, {c, f}, {c, g}, {d, e}, {d, f}, {d, g}, {e, f}, {e, g}, {f, g}, {a, c,
d}, {b, f, g}, {c, d, e}, {c, d, f}, {c, d, g}, {c, e, f}, {c, e, g}, {c, f, g},
{d, e, f}, {d, e, g}, {d, f, g}, {e, f, g}, {a, c, d, e}, {a, c, d, f},
{a, c, d, g}, {b, c, f, g}, {b, d, f, g}, {b, e, f, g}, {c, d, e, f}, {c, d, e,
g}, {c, d, f, g}, {c, e, f, g}, {d, e, f, g}, {a, c, d, e, f}, {a, c, d, e, g},
{a, c, d, f, g}, {b, c, d, f, g}, {b, c, e, f, g}, {b, d, e, f, g}, {c, d, e, f,
g}, {a, c, d, e, f, g}, {b, c, d, e, f, g}, X}; S*C(X, τ)={ϕ, {e}, {c, d},
{f, g}, {a, c, d}, {b, f, g}, {c, d, e}, {e, f, g}, {a, c, d, e}, {b, e, f, g},
{c, d, f, g}, {a, c, d, f, g}, {b, c, d, f, g},
{c, d, e, f, g}, {a, c, d, e, f, g}, {b, c, d, e, f, g}, X}.
Let A={e, f, g}. Then Cl(A)=s*Cl(A)=sCl(A)=A.
Let B={b, c, d, f, g}. Then Cl(B)={b, c, d, e, f, g};
s*Cl(B)=sCl(B)={b, c, d, f, g}.
Here B=sCl(B)=s*Cl(B)⊊Cl(B).
Let C={f}. Then sCl(C)={f}; s*Cl(C)=Cl(C)={f, g}. Here
C=sCl(C)⊊s*Cl(C)=Cl(C).
Let D={a, c, e}. Then sCl(D)=s*Cl(D)=Cl(D)={a, c, d, e}. Here
D ⊊sCl(D)=s*Cl(D)=Cl(D)
Let E={b, d, f, g}. Then sCl(E)={b, d, f, g}; s*Cl(E)={b, c, d, f,
g}; Cl(E)={ b, c, d, e, f, g}.
Here E=sCl(E)⊊s*Cl(E)⊊Cl(E).
Let F={a, f, g}. Then sCl(F)=s*Cl(F)={a, c, d, f, g}; Cl(F)={ a, c,
d, e, f, g}.
Here F⊊sCl(F)=s*Cl(F)⊊Cl(F).
Let G={b, c, e, g}. Then sCl(G)={b, c, e, f, g}; s*Cl(G)=Cl(G)={b,
c, d, e, f, g}.
Here G⊊sCl(G)⊊s*Cl(G)=Cl(G).
Let H={a, g}. Then sCl(H)={a, c, d, g}; s*Cl(H)={a, c, d, f, g};
Cl(H)={ a, c, d, e, f, g}.
Here H⊊sCl(H)⊊s*Cl(H)⊊Cl(H).
Remark 3.43: The inclusions in (vii) and (viii) of Theorem
3.40, may be strict and equality may also hold. This can be
seen from the following examples.
Example 3.44: Consider the space (X, τ) in Example 3.42.
Let A={a, d} and B={d, e} then A∪B={a, d, e} and A⋂B={d}.
s*Cl(A)={a, c, d}; s*Cl(B)={c, d, e}; s*Cl(A∪B)={a, c, d, e};
s*Cl(A⋂B)={c, d}.
Here s*Cl (A∪B)=s*Cl (A)∪s*Cl(B) and s*Cl(A⋂B)=s*Cl
(A)⋂s*Cl (B).
Let C={b, f } and D={a, c} then C∪D={a, b, c, f}; s*Cl (C)={b, f,
g}; s*Cl (D)={a, c, d};
s*Cl(C∪D)=X ; s*Cl(C)∪s*Cl(D)={a, b, c, d, f, g}. Here
s*Cl(C)∪s*Cl(D)⊊s*Cl(C∪D).
Let E={a, e} and F={c, e, f} then E⋂F={e}; s*Cl(E)={a, c, d, e};
s*Cl(F)={c, d, e, f, g}; s*Cl(E⋂F)={e}; s*Cl(E)⋂s*Cl(F)={c, d,
e}.Here s*Cl(E⋂F)⊊ s*Cl(E)⋂s*Cl(F).
Theorem 3.45: Let ß be a collection of subsets in (X, τ)
satisfying (i) 𝓕⊆ß where 𝓕 is the collection of closed sets in
(X, τ) (ii) If B∈ß and D⊆X such that Int*(B)⊆D⊆B implies
D∈ß. Then S*C(X, τ)⊆ß. Thus S*C(X, τ) is the smallest
collection satisfying the (i) and (ii).
Proof: Let A∈S*C(X, τ).Then by Theorem 3.7, there is a
closed set F in (X, τ) such that Int*(F)⊆A⊆F. By
(i), F∈ß. By (ii), A∈ß. Hence S*C(X, τ)⊆ß. By using Theorem
3.18 and Theorem 3.10, S*C(X, τ) satisfies (i) and (ii).Thus
S*C(X, τ) is the smallest collection satisfying (i) and (ii).
Theorem 3.46: Let A⊆X and let x∈X. Then x∈s*Cl(A) if and
only if every semi*-open set in X containing x intersects A.
Proof: Sufficiency. Suppose x∉s*Cl(A). Then X\s*Cl(A) is a
semi*-open set containing x that does not intersect A.
Necessity. Suppose U is a semi*-open set containing x that
does not intersect A. Then X\U is a semi*-closed set
containing A. Therefore s*Cl(A)⊆X\U. Hence
x∉s*Cl(A). Thus x∉s*Cl(A) if and only if there is a semi*open set containing x that does not intersect A. This proves
the theorem.
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Corollary 3.47: Let A be a subset of X. s*Cl(A) is the set of
all semi*-adherent points of A.
Proof: Follows from Definition 3.3.
Theorem 3.48: If A is a subset of X, then
(i) s*Cl(X\A)=X\s*Int(A).
(ii) s*Int(X\A)=X\s*Cl(A).
Proof: (i) Let x∈X\s*Int(A).Then x∉s*Int(A).This implies
that x does not belong to any semi*-open subset of A. Let F
be a semi*-closed set containing X\A. Then X\F is a semi*open set contained in A. Therefore x∉X\F and so x∈F.
Hence x∈s*Cl(X\A). Therefore X\s*Int(A)⊆ s*Cl(X\A). On
the other hand, let x∈s*Cl(X\A). Then x belongs to every
semi*-closed set containing X\A. Hence x does not belong
to any semi*-open subset of A. That is x∉s*Int(A). Then
x∈X\s*Int(A).This proves (i).
(ii) can be proved by replacing A by X\A in (i) and using set
theoretic properties.
Theorem 3.49: If A⊆X, then s*Cl(A)=A∪Ds*[A].
Proof: By definition, A⊆s*Cl(A). If x∈Ds*[A], then by
Theorem 3.47 and Definition 3.5, x∈s*Cl(A). Thus A∪
Ds*[A]⊆s*Cl(A). On the other hand, let x∈s*Cl(A). If x∈A,
then x∈A∪Ds*[A]. Suppose x∉A. We claim that x is a semi*limit point of A. Let U be a semi*-open set containing x.
Then U intersects A. Otherwise X\U is a semi*-closed set
containing A and since x∈s*Cl(A), we have x∈X\U. This is a
contradiction to x∈U. Hence x∈Ds*[A]. Therefore
s*Cl(A)⊆A∪Ds*[A]. Hence the theorem.
Corollary 3.50: A subset A of X is semi*-closed if and only
if A contains all its semi*-limit points. That is, A is semi*closed if and only if Ds*[A]⊆A.
Proof: Follows from Theorem 3.17 and Theorem 3.49.
Theorem 3.51: If A is a subset of a topological space X,
(i) s*Cl(A)=A∪Int*(Cl(A)).
(ii) s*Int(A)=A⋂Cl*(Int(A)).
Proof:
Int*(Cl(A∪Int*(Cl(A))))=Int*(Cl(A)∪Cl(Int*(Cl(A))))
=Int*(Cl(A))⊆A∪Int*(Cl(A)). Then by Theorem 3.9,
A∪Int*(Cl(A)) is a semi*-closed set containing A. Hence
s*Cl(A) ⊆A∪Int*(Cl(A)). Also since s*Cl(A) is semi*-closed ,
by
invoking
Theorem
3.9
again,
we
get
Int*(Cl(A))⊆Int*(Cl(s*Cl(A)))⊆s*Cl(A).
Hence
A∪Int*(Cl(A))⊆s* Cl(A) which proves (i).
(ii) follows from (i) above by replacing A by X\A and
applying Theorem 3.48.
Theorem 3.52: If a subset A of a space X is nowhere dense,
then Int(s*Cl(A))=.
Proof:
Follows from
Int(Cl(A))=
and the fact
Int(s*Cl(A))⊆Int(Cl(A)).
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