J. Comp. & Math. Sci. Vol.4 (4), 202-209 (2013)
On µˆ -T0, µˆ -T1, µˆ -T2, µˆ -R0, µˆ -R1 Spaces S. PIOUS MISSIER1 and E. SUCILA2 1
Department of Mathematics, V.O. Chidambaram College, Tuticorin, INDIA. 2 Department of Mathematics, G. Venkataswamy Naidu College, Kovilpatti, INDIA. (Received on: June 5, 2013) ABSTRACT M.K.R.S. Veerakumar introduced µ-closed sets in topological space. Using µ-closed sets recently S.Pious Missier and E.Sucila have defined and studied the notions of
µˆ -closed and µˆ -opensets
in topological spaces. In this paper, we introduce and investigate
µˆ -Ti,
i = 0,1,2 and
µˆ -Ri,
i = 1,2 using these
µˆ -opensets
in
topological spaces.
µˆ -Ti (i = 0,1,2) spaces, µˆ -Ri(i = 1,2) spaces, semi ˆ -open map, µˆ -irresolute map. symmetric, always µ Keywords:
2000 Mathematics Subject code Classification: 54B40, 54C05, 54D05.
1. INTRODUCTION In 1963, the concept of semiopen sets in topology was introduced by N. Levine3. In4, N. Levine generalized the concept of closed set to generalized closed set. The concept of semi-Ti, i = 0,1,2 spaces has been defined by S.N. Maheshwari and R.Prasad in5, which is weaker than Ti, i = 0,1,2 spaces. The notion of µ-closed sets was introduced by M.K.R.S. Veerakumar11. ˆ -closed Recently the authors introduced8 µ
sets in topological spaces using µ-closed set. In this paper, we introduce and investigate µˆ -Ti, i = 1,2,3 and µˆ -Ri, i = 1,2 using µˆ open sets in topology. Further, we study their basic properties and preservation theorems of these new spaces. 2. PRELIMINARIES Throughout this paper, by (X, τ), (Y, σ) (or simply X and Y), we always mean topological spaces on which no separations
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axioms are assumed unless explicity stated. Let A be a subset of X. The interior and closure of A in X are denoted by int(A) and cl(A) respectively. The following definitions and results are useful in the sequel.
ˆ -closed sets The intersection of all µ containing a subset A of X is called µˆ -closure of A is denoted by µˆ cl(A)9.
Definition: 2.1 A subset A of a topological space (X,τ) is called:
A space X is said to be : 1. semi-T05 if for each pair of distinct points in X, there is an semi-openset containing one of the points but not the other. 2. semi-T15 if for each pair of distinct points x and y of X, there exists semiopen sets U and V containing x and y respectively such that y ∉ U and x ∉ V. 3. semi-T25 if for each pair of distinct points x and y of X, there exist disjoint semi-open sets U and V containing x and y respectively. 4. semi-R02 if for each semi-openset G in X and x ∈ G such that scl({x}) ⊂ G. 5. semi-R12 if for x, y ∈ X with scl({x}) ≠ scl({y}), there exist disjoint semi-open sets U and V such that scl({x}) ⊂ U and scl({y}) ⊂ V.
1. semiopen3 if A ⊆ cl(int(A)) 2. α-open7 if A ⊆ int(cl(int(A))) The complement of semi-open (resp. α-open) set is called semi closed (resp. αclosed). The intersection of all semiclosed (resp. α-closed) sets containing a subset A of X is called semiclosure (resp. α-closure) of A is denoted by scl(A) (resp. αcl(A)).
Definition : 2.3
Definition: 2.2 A subset A of a topological space (X, τ) is called 1. gα*-closed6 if αcl(A) ⊆ int(U) whenever A ⊆ U and U is α-open in (X, τ). The complement of gα*-closed set is called gα*-open. 2. µ-closed set [11] if cl(A) ⊆ U whenever A ⊆ U and U is gα*-open in (X, τ). The complement of µ-closed set is called µ-openset. ˆ -closed8 if scl(A) ⊆ U whenever A ⊆ 3. µ U and U is µ-open in (X, τ). The ˆ -closed set is called µˆ complement of µ -openset.
Definition: 2.4 A mapping f : X → Y is said to be always α-open1 if the image of every αopenset of X is an α-openset in Y.
µˆ -closed subsets of X ˆ c(X, τ) and the class of µˆ is denoted by µ ˆ o(X, τ)8. open subsets of X is denoted by µ
Definition: 2.6 ˆA mapping f : X → Y is called µ irresolute [10] if the inverse image of each µˆ -openset in Y is an µˆ -openset in X.
The class of
Definition: 2.5 ˆ A mapping f : X → Y is called an µ 9 -continuous if the inverse image of each ˆ -openset in X. openset in Y is an µ
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3. ON
µˆ -T0 SPACES
We definition.
ˆ cl({y}) but x ∉ µˆ cl({y}) as contain y, y ∈ µ
introduce
the
following
Definition 3.1 A topological space (X, τ) is said to ˆ -T0, if for each pair of distinct points x, be µ
ˆ -openset containing y of X, there exists an µ one point but not the other. Clearly every ˆ -T0, since every semi-T0 space is µ ˆ -open in X. semiopen set in X is µ However the converse is not true in general as shown by the Example 3.2.
Example: 3.2 Let X = {a, b, c} and τ = {X, ϕ, {a}, ˆ -T0 but not semi– T0, {b, c}}. Here X is µ since there is no semiopen set containing b but not containing c. The following Theorem 3.3 ˆ -T0 spaces. characterizes µ Theorem: 3.3 A space X is an only if distinct.
µˆ -T0 space if and
µˆ -closures of distinct points are
ˆ cl({x}) ≠ µˆ cl({y}). x ∉ X – G. Therefore µ Conversely, suppose that for any ˆ cl({x}) ≠ pair of distinct points x, y ∈ X, µ µˆ cl({y}). Then there exists atleast one point ˆ cl({x}) but z ∈ X such that z ∈ µ ˆ cl({y}). We claim that x ∉ µˆ cl({y}). z ∉µ ˆ cl({y}), then µˆ cl({x}) ⊂ µˆ If x ∈ µ ˆ cl({y}), which is a cl({y}). So z ∈ µ ˆ cl({y}). Now contradiction. Hence x ∉ µ ˆ cl({y}) ⇒ x ∈ X- µˆ cl({y}), which is x ∉µ ˆ -openset in X containing x but not y. an µ ˆ -T0 space. Hence x is an µ Definition: 3.4 A mapping f : X → Y is said to be ˆ -open if the image of every always µ
µˆ -openset of X is µˆ -open in Y.
Proposition: 3.5 ˆ -T0 The property of a space being µ is preserved under one-one, onto and always µˆ -open mapping. Proof:
Proof: Let x, y ∈ X with x ≠ y and X be an µˆ -T0 space. We shall show that µˆ cl({x}) ≠
µˆ cl({y}). Since X is µˆ -T0 space, there ˆ -open set G such that x ∈ G and exists an µ y ∉ G. Also x ∉ X – G and y ∈ X – G where ˆ -closed set in X. Since µˆ cl({y}) X – G is µ is the intersection of all
204
µˆ -closed sets which
ˆ -T0 space and Y be a Let X be a µ topological space. Let f : X → Y be a oneˆ -open mapping from X one, onto, always µ to Y. Let u, v ∈ Y with u ≠ v. Since f is oneone onto, there exists distinct points x, y ∈ X such that f(x) = u, f(y) = v. Since X ˆ ˆ is an µ -T0 space, there exists µ -openset G in X such that x ∈ G and y ∉ G. Since ˆ -open, f(G) is an µˆ -openset in f is always µ
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Y containing f(x) = u but not containing f(y) ˆ -open set f(G) in = v. Thus there exists an µ Y such that u ∈ f(G) but v ∉ f(G) and hence ˆ -T0 space. Y is an µ 4. ON
µˆ -T1 SPACES
In this section, we define spaces and discuss its properties.
µˆ -T1
Definition: 4.1 ˆ -T1, if for A space X is said to be µ each pair of distinct points x, y of X, there ˆ -opensets, one containing x exists a pair of µ but not y and the other containing y but not x. It is easy to verify the following. ˆ -T1 space. (i) Every semi–T1 space is an µ
ˆ -T1 space is an µˆ -T0 space. (ii) Every µ The converses of the above two statements are not true as shown by the examples. Example: 4.2 Let X = {a, b, c} and τ = {X, ϕ, ˆ -T1 space but {a}, {b, c}}. Then X is an µ not semi-T1 space, since there is no semi openset containing b but not containing c. Example: 4.3 Let X = {a, b, c} and τ = {X, ϕ, {a}, ˆ -T0 space but not an {a, c}}. Then, X is an µ
Proof : Let x, y be two distinct points of X ˆ -closed in X. such that {x} and {y} are µ
ˆ -open in X Then X – {x} and X – {y} are µ such that y ∈ X – {x} but x ∉ X – {x} and x ∈ X–{y} but y ∉ X – {y}. Hence, X is an µˆ -T1 space. Theorem: 4.5 If {x} is
µˆ -closed for each x in X
and scl({x}) is µ-closed for each x in X then a space X is semi-symmetric. Proof: Suppose x ∈ scl({y}) and y ∉ scl({x}). Since y ∈ X – scl({x}) and scl({x}) is µ-closed implies X-scl({x}) is µ-open also {y} is
µˆ -closed by definition of µˆ -closed,
scl({y}) ⊂ X – scl({x}). Thus x ∈ X – scl({x}), a contradiction. Next, we have the following invariant properties. Theorem: 4.6
µˆ -irresolute, ˆ -T1, then X is µˆ -T1. injective map. If Y is µ Let f : X → Y be an
Proof:
µˆ -T1. Let x, y ∈
µˆ -T1, since for distinct point a and b of X ˆ -opensets containing there is no pair of µ
X be such that x ≠ y. Then there exists a pair
one point but not the other.
of
Theorem: 4.4 A space X is an
f(y)∈V and f(x)∉V, f(y)∉U. Then x ∈ f-1(U), y ∈ f-1(V), x ∉ f-1(V) and y ∉ f-
µˆ -T1 space if {x} is
µˆ -closed in X for every x ∈ X.
Assume that Y is
1
µˆ -opensets U, V in Y such that f(x)∈U,
(U). Since f is
µˆ -irresolute, X is µˆ -T1.
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5. ON
µˆ -T2 SPACES
Definition: 5.1 ˆ -T2, if for A space X is said to be µ each pair of distinct points x, y of X, there ˆ -opensets U and V such that exist disjoint µ x ∈ U and y ∈ V. Clearly the following holds. ˆ -T2 Every semi-T2 space is an µ space. However the converse is not true in general as shown by the example. Example: 5.2 The space defined in Example 4.2 is ˆ -T2 space but not an semi-T2 space, an µ since the semi-openset {b, c} containing both b and c of X. ˆ -T2 space is µˆ -T1 Also every µ space. We have the following invariant properties. Theorem: 5.3 Let f : X → Y be an
ˆ cl(U) / U is (c) For each x ∈ X , ∩ { µ open in X and x ∈ U} = {x}. (d) The diagonal ∆ = {(x, x)/ x ∈ X} is closed in X × X.
206
µˆ µˆ -
Proof:
ˆ(a) ⇒ (b) : Assume that the space X is µ T2. Let x, y ∈ X such that y ≠ x. Then ˆ -opensets U and V in there are disjoint µ X such that x ∈ U and y ∈ V. Clearly X ˆ -closed such that µˆ cl(U) ⊂ X – – V is µ ˆ cl(U). V, y ∉ X – V and therefore y ∉ µ (b) ⇒ (c) : Assume that for each y ≠ x, there ˆ -openset U such that exists an µ x∈ ˆ cl(U). So, y ∉ ∩ { µˆ cl(U) / U and y ∉ µ
µˆ -open in X and x ∈ U} = {x}. ˆ -open (c) ⇒ (d) : We claim that X - ∆ is µ U is
in X × X. Let (x, y) ∉ ∆. Then y ≠ x. ˆ cl(U)/ U is µˆ -open in X Since ∩ { µ
µˆ -irresolute ˆ -T2, then X is µˆ -T2. and injective. If Y is µ
ˆand x ∈U} = {x}, there is some µ openset U in X with x ∈ U ˆ cl(U)). Since U ∩ (X - µˆ and y ∉ µ
Proof : Similar to Theorem 4.6. The following theorem 5.4 ˆ -T2 spaces. characterizations of µ
cl(U)) = ϕ, U × (X -
Theorem: 5.4 In a space X, statements are equivalent. ˆ -T2. (a) X is µ
the
gives
following
ˆ(b) For each y ≠ x ∈ X, there exists an µ openset U such that x ∈ U and ˆ cl(U). y ∉µ
µˆ cl(U)) is an µˆ ˆ open set such that (x, y) ∈ U × (X - µ ˆ -closed in cl(U)) ⊂ X – ∆. Hence ∆ is µ X × X. (d) ⇒ (a) : If y ≠ x, then (x, y) ∉ ∆ and thus ˆ -open sets U and V such there exist µ that (x, y) ∈ U × V and (U × V) ∩ ∆ = ˆ -open sets U and V, ϕ. Thus, for the µ we have x ∈ U, y ∈ V and U ∩ V = ϕ. ˆ -T2. Hence X is µ
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µˆ -R0 space if µˆ cl({x}) ⊂ U whenever ˆ -open set and x ∈ U. U is µ ˆ -R1 space if for x, y ∈ X with µˆ (ii) µ ˆ cl({y}), then there exist cl({x}) ≠ µ ˆ -open sets U and V such that disjoint µ µˆ cl({x}) ⊂ U and µˆ cl({y}) ⊂ V.
Remark: 5.5 From the above Propositions and Examples, we have the following diagram. Here A → B represents A implies B but not conversely. In 1975, S. N. Maheshwari and R. Prasad5 have shown that the following implications hold.
(i)
T2
semi-T2
T1
semi-T1
Remark: 6.2 Every space.
T0
Proposition: 6.3 A topological space (X, τ) is
semi-T0
µˆ -R0
ˆ c(X, τ), x ∉ F implies then for any F ∈ µ
Now the authors have shown that the following implications hold. T2
µˆ -R1 space is an µˆ -R0
µˆ -T2
semi-T2
F ⊂ U and x ∉ U for some U ∈ Proof:
µˆ o(X, τ).
ˆ c(X, τ) and x ∉ Suppose that F ∈ µ
F. Then x ∈ X – F also. X – F is
µˆ -open.
µˆ -R0, µˆ cl({x}) ⊂ X – F. Set ˆ cl({x}) then U ∈ µˆ o(X, τ), U = X - µ
Since (X, τ) is T1
T0
µˆ -T1
semi-T1
µˆ -T0
semi-T0
ˆ -R0 SPACES AND 6. ON µ SPACES
µˆ -R1
In this section, we define
µˆ -R0
ˆ -R1 spaces and study some of spaces and µ the properties. Definition: 6.1 A space X is called an
F ⊂ U and x ∉ U. Proposition: 6.4 ˆ -R0 A topological space (X, τ) is µ then for any nonempty set A and ˆ o(X, τ) such that A ∩ G ≠ ϕ, there G ∈µ
ˆ c(X, τ) such that A ∩ F ≠ ϕ and exists F ∈ µ F ⊂ G. Proof: Let A be a nonempty set of X and G ˆ o(X, τ) such that A ∩ G ≠ ϕ. There ∈ µ exists x ∈ A ∩ G. Since x ∈ G ∈
µˆ o(X, τ),
µˆ cl({x}) ⊂ G. Set F = µˆ cl({x}), then F ∈ µˆ c(X, τ), F ⊂ G and A ∩ F ≠ ϕ.
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Proposition: 6.5 A topological space (X, τ) is
µˆ -R0 ˆ space if and only if for any x and y in X, µ ˆ cl({y}) implies µˆ cl({x}) ∩ µˆ cl({x}) ≠ µ
Proof : (a) ⇒ (b). Suppose X is
µˆ -R0. Let x ∈ µˆ ˆ -openset such cl({y}) and U be any µ
that y ∈ U. By hypothesis, x ∈ U. ˆ -openset containing Therefore, every µ
cl({y}) = ϕ.
ˆ cl({x}). y contains x. Hence y ∈ µ
Proof: Necessity: Suppose that (X, τ) is
µˆ -R0 and ˆ cl({x}) ≠ µˆ cl({y}). x, y ∈ X such that µ ˆ cl({x}) such that z Then, there exists z ∈ µ ˆ cl({y}) (or z ∈ µˆ cl({y}) such that z ∉ µˆ ∉µ ˆ o(X, τ) such that cl({x}). There exists V ∈ µ y ∉ V and z ∈ V, hence x ∈ V. Therefore ˆ cl({y}). Thus x ∈ X - µˆ we have x ∉ µ
µˆ o(X, τ), sinceX is µˆ -R0, µˆ ˆ cl({y}) and µˆ cl({x}) ∩ µˆ cl({x}) ⊂ X - µ cl({y}) ∈
cl({y}) = ϕ. The proof for otherwise is similar.
ˆ o(X, τ) and let x ∈ Sufficiency: Let V ∈ µ ˆ cl({x}) ⊂ V. V. Now we will show that µ Let y ∉ V(ie) y ∈ X – V. Then x ≠ y and x ˆ cl({y}). This shows that µˆ cl({x}) ≠ µˆ ∉ µ cl({y}). By assumption cl({y}) = ϕ. Hence y ∉
µˆ cl({x}) ∩ µˆ
µˆ cl({x}). Therefore
µˆ cl({x}) ⊂ V. Theorem : 6.6 The following equivalent for a space X. (a) X is an
properties
208
are
µˆ - R0 space.
ˆ cl({y}) if and only if y ∈ µˆ cl({x}) (b) x ∈ µ for points x and y in X.
µˆ -openset and x ∈ ˆ cl({y})and V. If y ∉ V, then x ∉ µ ˆ cl({x}). This implies that hence y ∉ µ µˆ cl({x}) ⊂ V. Hence X is µˆ -R0.
(b) ⇒ (a). Let V be an
Theorem: 6.7 If a space X is it is
µˆ -R1 and µˆ -T0 then
µˆ -T2.
Proof : Let x, y be any two distinct points in ˆ -T0 which X. By Theorem 3.3, X is µ
µˆ -closures of distinct points ˆ -R1, there exist are distinct. Since X is µ ˆ -open sets U and V such that µˆ disjoint µ ˆ cl({y}) ⊂ V and so x ∈ U cl({x}) ⊂ U and µ ˆ -T2 space. and y ∈ V. Hence X is an µ implies that
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4. N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo 19(2), 89 – 96 (1970). 5. S.N. Maheshwari and R. Prasad, Some new separation axioms, Ann. Soc. Sci. Bruxelles Ser.I. 89, 395 – 402 (1975). 6. H. Maki, R. Devi and K. Balachandran, Generalized α-closed sets in topology, Bull Fukuoka Univ. Ed. Part III 42, 13 – 21 (1993). 7. O. Njastad, On some classes of nearly opensets, Pacific J. Math., 15, 961–970 (1965). ˆ8. S. Pious Missier and E. Sucila, On µ closed sets in topological spaces.
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