BETWEEN CLOSED SETS AND g-CLOSED SETS

M.K.R.S. VEERA KUMAR*

Department of Mathematics, Nagarjuna University, Nagarjuna Nagar-522 510,

GUNTUR, A.P., INDIA

* Author's Address for correspondence : J.K.C.College, GUNTUR -522 006, A.P.

1991 AMS classification : 54 A 05, 54 D 10.

Key words and phrases : g-closed sets; g*-closed sets; g*-continuous maps; g*-irresolute maps; T_1/2* spaces; *T_1/2 spaces; _aT_c spaces and T_c spaces.

Abstract: In this paper we introduce a new class of sets namely, g*-closed sets, which settled properly in between the class of closed sets and the class of generalized closed sets. Applying these sets, the author introduced four new class of spaces, namely T_1/2 * spaces, *T_1/2 spaces (both classes contain the class of T_1/2 spaces), _aT_c spaces and T_c spaces. The class of T_c spaces is properly placed in between the class of T_b spaces and the class of T_d spaces. It is shown that dual of the class of T_1/2* spaces to the class of _aT_b spaces is the class of _aT_c spaces and the dual of the class of *T_1/2 spaces to the class of T_1/2 spaces is the class of T_1/2* spaces and also that the dual of the class of T_d spaces to the class of T_c spaces is the class of _aT_c spaces. Further we introduce g*-continuous maps and g*-irresolute maps.

§1. INTRODUCTION

..........Levine [16] introduced the class of g-closed sets, a super class of closed sets in 1970. N.V.Velicko [26] defined two subclasses of closed sets namely, d-closed sets and q-closed sets in 1968... Maki et.al. [18] defined .ag-closed sets and a**g-closed sets in 1994. S.P.Arya and T.Nour [4] defined gs-closed sets in 1990, which were used for characterizing s-normal spaces. Dontchev [11], Gnanambal [15] and Palaniappan and Rao [24] introduced gsp-closed sets, gpr-closed sets and r-g-closed sets respectively. We introduce a new class of sets (using new technique) called g*-closed sets, which is properly placed in between the class of closed sets and the class of g-closed sets. We also showed that this new class is properly contained in the class of ag-closed sets, the class of gs-closed sets, the class of gsp-closed sets, the class of gpr-closed sets, the class of r-g-closed sets and the class of a**g-closed sets and properly contains the class of d-closed sets and the class of q-closed sets.

..........Levine [17], Mashhour et.al.[21] Njastad [23] and Abd-El-Monsef et.al. [1] introduced semi-open sets, preopen sets, a-sets and b-sets respectively. Andrijevic [2] called b-sets as semi-preopen sets. The complement of a semi-open (resp.preopen, a-open, semi-preopen) set is called a semi-closed (resp.preclosed, a-closed, semi-preclosed) set. Maki et.al. [19] and Bhattacharya and Lahiri [6] introduced and studied ga-closed sets and sg-closed sets respectively. We proved that g*-closedness is independent from semi-closedness, pre-closedness, a-closedness, semi-preclosedness, sg-closedness and ga-closedness.Applying g*-closed sets, four new spaces namely, T_1/2* spaces, *T_1/2 spaces, Tc spaces and _aT_c spaces are introduced. Levine [16], Devi et.al.[8] and Devi et.al.[7] introduced T_1/2 spaces, T_b (T_d) spaces, and _aT_b( _aT_d) spaces respectively. T_c(_aT_c) is properly placed in between T_b( _aT_b) and T_d(_aT_d). It is shown that T_1/2* (*T_1/2) is strictly weaker than T_b, _aT_b and T_1/2 ( _aT_c) and T_1/2. We also found that _aT_c ness is the dual of T_1/2* ness to _aT_b ness and *T_1/2 ness is the dual of T_1/2* ness to T_1/2 ness. We also get a tri-decomposition for T_b ness.

§ 2. PRELIMINARIES

..........Throughout this paper (X, t), (Y, s) and (Z, h) represent non-empty topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space (X, t), cl(A) and int(A) denote the closure and the interior of A respectively. The power set of X is denoted by P(X).

..........DEFINITION 2.01 - A subset A of a topological space (X, t) is called

(1) a pre-open set [21] if A Í int(cl(A)) and a preclosed set if cl(int(A)) Í A.

(2) a semi-open set [17] if A Í cl(int(A)) and a semi-closed set if int (cl(A)) Í A.

(3) an a-open set [23] if A Í int(cl(int(A))) and an a-closed set [22] if cl (int(cl(A))) Í A.

(4) a semi-preopen set [2] (= b-open[1]) if A Í cl(int(cl(A))) and a semi preclosed set [2] (= b-closed [1]) if int(cl(int(A))) .....Í A.

(5) a d-closed set [26] if A = cld(A), where cl_d(A) = {xÎX : int(cl(U)) Ç A ¹ f,UÎt and x ÎU}.

(6) a q-closed set [26] if A = clq(A), where cl_q(A) = {xÎX : cl(U)ÇA ¹ f, UÎt and x ÎU}.

..........The class of all closed (resp.semi-preclosed, a-open) subsets of a space (X,t) is denoted by C(X,t) (resp.SPC(X,t),ta). The intersection of all semi-closed (resp.pre-closed, semi-preclosed and a-closed) sets containing a subset A of (X,t) is called the semi-closure (resp.pre-closure, semi-pre-closure and a-closure) of A and is denoted by scl(A)(resp.pcl(A),spcl(A) and acl(A)).

..........DEFINITION 2.02 - A subset A of a topological space (X, t) is called

(1) a generalized closed set (briefly g-closed)[16] if cl(A) Í U whenever A Í U and U is open in (X, t).

(2) a semi-generalized closed set (briefly sg-closed) [6] if scl(A) Í U whenever A Í U and U is semi-open in (X, t).

(3) a generalized semi-closed set (briefly gs-closed) [4]if scl(A) Í U whenever A Í U and U is open in (X, t).

(4) a generalized a-closed set (briefly ga-closed) [19] if acl(A) Í U whenever A Í U and U is a-open in (X, t).

(5) an a-generalized closed set (briefly ag-closed) [18] if acl(A) Í U whenever A Í U and U is open in (X, t).

(6) an a**-generalized closed set (briefly a**g-closed) [18] if acl(A) Í int(cl(U)) whenever A Í U and U is open in (X, t).

(7) a ga*-closed set [19] if acl(A) Í int(U) whenever A Í U and U is a-open in (X, t).

(8) a generalized semi-preclosed set (briefly gsp-closed [11] if spcl(A) Í U whenever A Í U and U is open in (X, t).

(9) a regular generalized closed set (briefly r-g-closed) [24] if cl(A) ÍU whenever A Í U and U is regular open in (X, t).

(10) a generalized preclosed set (briefly gp-closed) [20] if pcl(A) Í U whenever A Í U and U is open in (X, t).

(11) a generalized preregular closed set (briefly gpr-closed) [15] if pcl(A) Í U whenever A Í U and U is regular open in .......(X, t).

(12) a q-generalized closed set (briefly q-g-closed) [13] if cl_q(A) Í U whenever A Í U and U is open in (X, t).

(13) a d-generalized closed set (briefly d-g-closed) [12] if cl_d(A) Í U whenever A Í U and U is open in (X, t).

..........The class of all g-closed sets (gsp-closed sets) of a space (X, t) is denoted by GC(X, t)(GSPC(X, t)).

..........DEFINITION 2.03 - A function f : (X, t) ® (Y, s) is called.

(1) a semi-continuous [17] if f ^-1(V) is a semi-open set of (X, t) for every open set V of (Y, s).

(2) a pre-continuous [21] if f ^-1(V) is a preclosed set of (X, t) for every closed set V of (Y, s).

(3) an a-continuous [22] if f ^-1(V) is an a-closed set of (X, t) for every closed set V of (Y, s).

(4) a b-continuous [1] if f ^-1(V) is a semi-preclosed set of (X, t) for every open set Vof (Y, s).

(5) a g-continuous [5] if f ^-1(V) is a g-closed set of (X, t) for every closed set V of (Y, s).

(6) an ag-continuous [15] if f ^-1(V) is an ag-closed set of (X, t) for every closed set V of (Y, s).

(7) a gs-continuous [9] if f ^-1(V) is a gs-closed set of (X, t) for every closed set V of (Y, s).

(8) a gsp-continuous [11] if f ^-1(V) is a gsp-closed set of (X, t) for every closed set Vof (Y, s).

(9) a rg-continuous [24] if f ^-1(V) is a rg-closed set of (X, t) for every closed set V of (Y, s).

(10) a gp-continuous [3] if f ^-1(V) is a gp-closed set of (X, t) for every closed set V of (Y, s).

(11) a gpr-continuous [15] if f ^-1(V) is a gpr-closed set of (X, t) for every closed set V of (Y, s).

(12) a gc-irresolute [5] if f ^-1(V) is a g-closed set of (X, t) for every g-closed set V of (Y, s).

(13) a gs-irresolute [9] if f ^-1(V) is a gs-closed set of (X, t) for every gs-closed set V of (Y, s).

(14) a ag-irresolute [7] if f ^-1(V) is a ag-closed set of (X, t) for every ag-closed set V of (Y, s).

..........DEFINITION 2.04 - A topological space (X, t) is said to be

(1) a T_1/2 space [16] if every g-closed set in it is closed.

(2) a semi-T_1/2 space [6] if every sg-closed set in it is semi-closed.

(3) a semi-pre-T_1/2 space [11] if every gsp-closed set in it is semi-preclosed.

(4) an _aT_1/2 space [19] if (X, t^a) is a T_1/2 space.

(5) an _aT₁ space [19] if (X, t^a) is a T₁ space.

(6) a T_b space [8] if every gs-closed set in it is closed.

(7) a T_d space [8] if every gs-closed set in it is g-closed.

(8) an _aT_d space [7] if every ag-closed set in it is g-closed.

(9) an _aT_b space [7] if every ag-closed set in it is closed.

(10) a T_3/4 space [12] if every d-generalized closed set in it is d-closed.

..........Further we call a function f : (X, t) ® (Y, s) as a**g-continuous if f ^-1(V) is an a**g-closed set of (X,t) whenever V is a closed set of (Y,s).

§ 3. BASIC PROPERTIES OF g*-CLOSED SETS

..........We introduce the following definition.

..........DEFINITION 3.01 - A subset A of (X, t) is called g*-closed set if cl(A) Í U whenever A Í U and U is g-open in (X, t).

..........The class of g*-closed subsets of (X, t). is denoted by G*C (X, t).

..........THEOREM 3.02 - Every closed set is a g*-closed set.

..........The following example supports that a g*-closed set need not be closed in general.

..........EXAMPLE 3.03 - Let X = {a, b, c} and t = {f, X, {a}, {a,c}}. Let A = {a, b}. A is a g*-closed set but not a closed set of (X, t).

..........So the class of g*-closed sets properly contains the class of closed sets. Next we show that the class of g*-closed sets is properly contained in the class of g-closed sets.

..........THEOREM 3.04 - Every g*-closed set is a g-closed set.

..........The converse of the above theorem is not true in general as it can be seen from the following example.

..........EXAMPLE 3.05 - Let X = {a, b, c}, t = {f, X, {a}, {b, c}} and B = {b}. B is not a g*-closed set since {b} is a g-open set of (X, t) such that B Í {b} but cl(B) = cl({b}) = {b, c} Í {b}. However B is a g-closed set of (X, t).

..........The following theorem shows that the class of g*-closed sets is properly contained in the class of ag-closed sets, the class of gs-closed sets, the class of gsp-closed sets, the class of gp-closed sets, the class of gpr-closed sets, the class of a**g-closed, the class of r-g-closed sets.

..........THEOREM 3.06 - Every g*-closed set is an ag-closed set and hence gs-closed, gsp-closed, gp-closed, gpr-closed set and also a**g-closed set and r-g-closed set but not conversely.

PROOF: Let A be a g*-closed set of (X, t). By the theorem 3.04, A is g-closed. By the implications (2.4) in Maki et.al.[18], A is ag-closed, and a**g-closed. From the investigations of Dontchev [11] and Gnanambal [15], we know that every g-closed set is gs-closed, gsp-closed, gp-closed, gpr-closed and r-g-closed. Therefore again by the Theorem 3.04, every g*-closed set is gs-closed, gsp-closed, gp-closed, gpr-closed and r-g-closed. The set B = {b} in the example 3.05, is ag-closed, gs-closed, gp-closed, gpr-closed, a**g-closed and r-g-closed but not a g*-closed set.

..........THEOREM 3.07 - Every d-closed set is a g*-closed set.

..........The converse of the above theorem is not true as we see the following example.

..........EXAMPLE 3.08 - Let X = {a, b, c}, t = {f, X, {a, b}} and D = {a, c}. D is not a d-closed set, in fact, it is not even a closed set. However D is g*-closed.

..........So the class of g*-closed sets not only contains the class of closed sets properly but also properly contains the class of d-closed sets. Next we see that the class of g*-closed sets properly contains the class q-closed sets also.

..........THEOREM 3.09 - Every q-closed set is a g*-closed set.

..........The following example shows that the converse of the above theorem is not true in general.

..........EXAMPLE 3.10 - Let X = {a, b, c}, t = {f, X, {a}, {a, b}, {a, c}} and E = {c}. Clearly E is closed and hence g*-closed. E is not a q-closed set of (X,).

..........REMARK 3.11 - g*-closedness is independent from a-closedness, semi-closedness preclosedness, semi-preclosedness, sg-closedness, ga-closedness and ga*-closedness.

PROOF : Let (X, t) be as in the example 3.03. Let B = {a, b} and D = {c}. B is g*-closed. B is neither a-closed nor semi-closed, in fact, it is not even a semi-preclosed set. Also C is not preclosed. D is a-closed and hence semi-closed, preclosed and semi-preclosed but it is not a g*-closed set. Also D is sg-closed, ga-closed and ga*-closed. Moreever B is not a sg-closed set.

..........REMARK 3.12 - If A and B are g*-closed sets, then A È B is also a g*-closed set.

PROOF: Follows from the fact that cl(A È B) = cl(A) È cl(B).

..........THEOREM 3.13 - If A is both g-open and g*-closed set of (X, t), then A is closed.

..........THEOREM 3.14 - A is a g*-closed set of (X, t) if and only if cl(A)-A does not contain any non-empty g-closed set.

PROOF : Necesssity-Let F be a g-closed set of (X, t) such that F Í cl(A)-A. Then A Í X-F. Since A is g*-closed and X-F is g-open, then cl(A) Í X-F. This implies F Í X-cl(A). So F Í (X-cl(A)) Ç (cl(A)-A) Í (X-cl(A)) Ç cl(A) = f. Therefore F = f.

Sufficiency- Suppose A is a subset of (X, t) such that cl(A)-A does not contain any non-empty g-closed set. Let U be a g-open set of (X, t) such that A Í U. If cl(A) Í U, then cl(A) Ç C(U) = f . Since cl(A) is a closed set, then by the Corollary 2.7 of [16], f = cl(A) Ç C(U) is a g-closed set of (X, t). Then f = cl(A) Ç C(U) Í cl(A)-A. So cl(A)-A contains a non-empty g-closed set. A contradiction. Therefore A is g*-closed.

The converse of the above theorem is not true as we see the following example.

..........THEOREM 3.15 - If A is a g*-closed set of (X,) such that A Í B Í cl(A), then B is also a g*-closed set of (X, t).

PROOF: Let U be a g-open set of (X, t) such that B Í U. Then A Í U. Since A is g*-closed, then cl(A) Í U. Now cl(B) Í cl(cl(A)) = cl(A) Í U. Therefore B is also a g*-closed set of (X, t).

..........3.16- Thus we have the following diagram.

[see my original paper]

§ 4. APPLICATIONS OF g*-CLOSED SETS

..........As applications of g*-closed sets, four new spaces namely, T_1/2* spaces, _aT_c spaces, *T_1/2 spaces and T_c spaces are introduced.

..........We introduce the following definition.

..........DEFINITION 4.01 - A space (X,) is called a T1/2* space if every g*-closed set is closed.

..........Levine [16] introduced T1/2 spaces as an application of g-closed sets. Dunham [14] proved that a space (X, t) is T_1/2 if and only if for each x Î X, {x} is either open or closed. R.Devi et.al. [8] proved that a space (X, t) is T_1/2 if and only if for each x Î X, {x} is either closed or semi-open and in a T_1/2space, every gs-closed set is semi-closed. Dontchev [10] showed that in a T_1/2 space, every ag-closed set is a-closed.Dontchev and Maki [13] proved that in a T_1/2 space, every q-g-closed set is closed.

..........We prove that the class of T_1/2* spaces properly contains the class of T_1/2 spaces.

..........THEOREM 4.02 - Every T_1/2 space is T_1/2* space.

..........A T_1/2*space need not be T_1/2 in general as it can be seen from the following example.

..........EXAMPLE 4.03 - Let X = {a, b, c} and t = {f, X, {a}}. G*C(X, t) = {{b, c}, f, X} = C(X, t). So (X, t) is not a T_1/2 space since {b} is a g-closed set but not a closed set of (X, t).

..........Bhattacharya and Lahiri [6] introduced semi-T1/2spaces as an application of sg-closed sets. Sundaram et.al.[25] proved that a space (X, t) is semi-T_1/2 if and only if for each x Î X, {x} is either semi-open or semi-closed. Dontchev [11] introduced semi-pre-T_1/2 spaces as an application of gsp-closed sets. He observed that semi-T_1/2ness and semi-pre-T_1/2ness are independent from each other and proved that semi-pre-T_1/2ness is the dual of semi-T_1/2ness to T_1/2 ness.

..........We show that T_1/2*ness is independent from semi-T_1/2ness and semi-pre-T_1/2ness.

..........REMARK 4.04 - T_1/2*ness and semi-T_1/2ness are independent as it can be seen from the next two examples.

..........EXAMPLE 4.05 - Let X and t be as in the example 3.03. (X, t) is not a T_1/2* space since {a, b} is a g*-closed set but not a closed set of (X, t). However (X, t) is a semi-T_1/2 space.

..........EXAMPLE 4.06 - Let X and t be as in the example 3.05. (X, t) is not a semi-T_1/2 space since {b} is neither a semi-open nor a semi-closed set of (X, t). However (X, t) is a T_1/2* space.

..........REMARK 4.07 - T_1/2* ness is independent from semi-pre-T_1/2 ness also as it can be seen from the next two examples.

..........EXAMPLE 4.08 - Let (X, t) be as in the example 3.08. (X, t) is not a T_1/2* space since {a, c} is a g*-closed set but not a closed set of (X, t). (X, t) is a semi-pre-T_1/2 space since SPGC(X, t) = P(X)-{{a, b}} = SPC(X, t).

..........EXAMPLE 4.09 - Let (X, t) be as in the example 4.03. (X, t) is not a semi-pre-T_1/2 space since {a, b} is a gsp-closed set but not a semi-preclosed set of (X, t). However (X, t) is a T_1/2*space.

..........Devi et.al.[8] and Devi et.al.[7] defined Tb spaces and aTb spaces respectively and showed that every Tb(_aT_b) space is a T_1/2 space.

..........Now we prove that every T_b(_aT_b) space is an _aT_b(T_1/2) space but the converses are not true.

..........THEOREM 4.10 - Every _aT_b space is T_1/2* space.

..........A T_1/2* space need not be an _aT_b as we see the next example.

..........EXAMPLE 4.11 - Let X and t be as in the example 4.03. Already we have seen that (X, t) is a T_1/2*space. (X,) is not an _aT_bspace since {b} is an ag-closed set but not a closed set of (X, t).

..........THEOREM 4.12 - Every T_b space is an _aT_b space but not conversely.

PROOF: The first assertion follows from the fact that every ag-closed set is also a gs-closed set. Let X = {a, b, c} and t = {f, X, {a}, {b}, {a, b}}. (X, t) is not a T_b space but an _aT_b space since aGC (X, t) = C(X, t).

..........THEOREM 4.13 - Every T_b space is a T_1/2* space.

..........The converse of the above theorem is not true in general as the following example supports.

..........EXAMPLE 4.14 -Let (X, t) be as in the example 4.03. Already we have seen that (X, t) is a T_1/2 * space. (X, t) is not a T_b space since it is not even a T_1/2 space.

..........The following theorem gives a characterization of T1/2* spaces.

..........THEOREM 4.15 - For a space (X, t), the following conditions are equivalent:

(1) (X, t) is a T_1/2* space.

(2) Every singleton of X is either g-closed or open.

PROOF: (1) Þ (2). Let x Î X and suppose {x} is not a g-closed set of (X, t). Then X - {x} is not g-open. This implies X is the only g-open set containing X-{x}. So X-{x} is a g*-closed set of (X, t). Since (X, t) is a T_1/2* space, then X-{x} is closed or equivalently {x} is open in (X, t).

(2) Þ (1). Let A be a g*-closed set of (X, t). Trivially A Í cl(A).Let x Î cl(A). By (2), {x} is either g-closed or open.

Case (i) - Suppose {x} is g-closed. If xÏ A, then cl(A)-A contains a non-empty g-closed set {x}. But this is not possible according to the theorem 3.14 as A is a g*-closed set. Therefore x Î A.

Case (ii) - Suppose {x} is open. Since x Î cl(A), then {x} Ç A ¹ f. So x ÎA. Therefore x ÎA. So in any case cl(A) Í A. Thus A = cl(A) or equivalently A is a closed set of (X, t).

..........We introduce the following definition.

..........DEFINITION 4.16 - A space (X, t) is called an _aT_c space if every ag-closed set of (X, t) is g*-closed.

..........We show that the class of _aT_c spaces properly contains the class of _aT_b spaces and is properly contained in the class of _aT_d spaces. We also show that the class of _aT_c spaces is the dual of the class of T_1/2* spaces to the class of _aT_b spaces. Moreover we prove that _aT_c ness and T_1/2*ness are independent from each other.

..........THEOREM 4.17 - Every _aT_b space is an _aT_c space but not conversely.

PROOF: Let (X, t) be an _aT_b space. Let A be an ag-closed set of (X, t). Since (X, t) is an _aT_b space, then A is closed. By the theorem 3.02, A is g*-closed. Therefore (X, t) is an _aT_c space. The space (X, t) in the example 3.08 is an _aT_c space but not an _aT_b space.

..........THEOREM 4.18 - Every _aT_c space is an _aT_d space but not conversely.

PROOF: Let (X, t) be an _aT_c space. Let A be ag-closed set of (X, t). Then A is g*-closed. By the theorem 3.04, A is g-closed. Therefore (X, t) is an _aT_d space. The space in the example 4.03 is an _aT_d space but not an _aT_c space.

..........THEOREM 4.19 - A space (X, t) is an _aT_b space if and only if it is _aT_c and T_1/2*.

PROOF: Necessity - Follows from the theorem 4.10 and 4.17.

Sufficiency - Suppose (X, t) is _aT_c and T_1/2*. Let A be an ag-closed set of (X, t). Since (X, t) is an _aT_c space, then A is a closed set of (X, t). Therefore (X, t) is an _aT_b space.

..........REMARK 4.20 - _aT_cness is independent from T_1/2*ness, as we see the next two examples.

..........EXAMPLE 4.21 - Let X and t be as in the example 3.08. (X, t) is not a T_1/2* space, as already we have seen in the example 4.08. However (X, t) is an _aT_c space.

..........EXAMPLE 4.22 - Let X and t be as in the example 3.05. G*C(X, t) = C(X, t). So (X, t) is a T1/2 * space. (X, t) is not an _aT_c space since {b} is an ag-closed set but not a g*-closed set of (X, t).

.........DEFINITION 4.23 - A subset A of a space (X,) is called a g*-open set if it's complement is a g*-closed set of (X, t).

..........THEOREM 4.24 - If (X, t) is an _aT_c space, then for each x Î X, {x} is either ag-closed or g*-open.

PROOF: Let x Î X and suppose that {x} is not an ag-closed set of (X, t). Then {x} is not a closed set since every closed set is an ag-closed set. So X-{x} is not an open set. Therefore X-{x} is an ag-closed set since X is the only open set which contains X-{x}. Since (X, t) is an aTc space, then X-{x} is a g*-closed set or equivalently {x} is g*-open.

..........REMARK 4.25 - The converse of the above theorem is not true as it can be seen from the following example.

..........EXAMPLE 4.26 - Let X and t be as in the example 3.05. The space (X, t) satisfies the conclusion of the above theorem 4.24. (X, t) is not an aTc space as already we have seen in the example 4.22.

..........We now introduce the following definition.

..........DEFINITION 4.27 - A space (X, t) is called a *T _1/2space if every g-closed set of (X, t) is a g*-closed set.

..........First we show that the class of *T_1/2 spaces properly contains the class of T_1/2 spaces and the class of _aT_c spaces.

..........THEOREM 4.28 - Every T_1/2 space is a *T_1/2 space but not conversely.

PROOF: Let (X, t) be a T_1/2 space. Let A be a g-closed set of (X, t). Since (X, t) is a T_1/2 space, then A is closed. By the theorem 3.02, A is a g*-closed set of (X, t).

..........Therefore (X, t) is a *T_1/2 space. The space (X, t) in the example 3.03 is a *T_1/2 space but not a T_1/2 space.

..........THEOREM 4.29 - Every _aT_c space is a *T_1/2 space but not conversely.

PROOF: Let (X, t) be an _aT_c space. Let A be a g-closed set of (X, t). Then A is also an ag-closed set. Since (X, t) is an _aT_c space, then A is a g*-closed set of (X, t). Therefore (X, t) is also a *T_1/2 space. The space in the example 3.03 is a *T1/2 space but not an _aT_c space.

..........Thus the class of *T_1/2 spaces is the dual of the class of T_1/2* spaces to the class of T_1/2 spaces. Now we show that *T_1/2ness and T_1/2*ness are independent from each other.

..........THEOREM 4.30 - A space (X, t) is a T_1/2 space if and only if it is *T_1/2 and T_1/2*.

PROOF: Necessity - Follows from the theorems 4.02 and 4.28.

Sufficiency - Suppose (X, t) is both T_1/2* and *T_1/2. Let A be a g-closed set of (X, t). Since (X, t) is *T_1/2, then A is g*-closed. Since (X, t) is a T_1/2* space, then A is a closed set of (X, t). Thus (X, t) is a T_1/2space.

..........REMARK 4.31 - T_1/2*ness and *T_1/2ness are independent as we see the next two examples.

..........EXAMPLE 4.32 - Let X and t be as in the example 3.05. (X, t) is not a *T_1/2 space since {b} is a g-closed set but not a g*-closed set of (X, t). However (X, t) is a T_1/2* space.

..........EXAMPLE 4.33 - Let X and t be as in the example 3.03. (X, t) is not a T_1/2* space as already we have seen in the example 4.05. However (X, t) is a *T_1/2space.

..........THEOREM 4.34 - If (X, t) is a *T_1/2space, then for each x Î X, {x} is either closed or g*-open.

PROOF: Suppose (X, t) is a *T_1/2 space. Let x ÎX and assume that {x} is not a closed set. Then X-{x} is not an open set. This implies X-{x} is a g-closed set since X is the only open set which contains X-{x}. Since (X, t) is a *T_1/2 space, then X-{x} is a g*-closed set or equivalently {x} is g*open.

..........Now we introduce the following definition.

..........DEFINITION 4.35 - A space (X, t) is called a T_c space if every gs-closed set of (X, t) is g*-closed.

..........We show that the class of T_c spaces properly contains the class of T_b spaces, the class of _aT_b spaces and is properly contained in the class of T_1/2 spaces, the class of _aT_c spaces, the class T_d spaces, the class of _aT_d spaces.

..........THEOREM 4.36 - Every T_b space is a T_c space but not conversely.

PROOF: Let (X, t) be a T_b space. Let A be a gs-closed set of (X, t). Since (X, t) is T_b, then A is closed set. By the theorem 3.02, A is also a g*-closed set. Therefore (X, t) is a Tc space. The space (X, t) in the example 3.08 is a T_c space but not a T_b space.

..........THEOREM 4.37 - Every T_b space is a T_d space but not conversely.

PROOF: Let (X, t) be a T_c space. Let A be a gs-closed set of (X, t). Since (X, t) is T_c, then A is g*-closed. By the theorem 3.04, A is also a g-closed set. Therefore (X, t) is also a T_d space. The space (X, t) in the example 3.05 supports the second assertion.

..........THEOREM 4.38 - Every Tc space is an _aT_c space.

PROOF: Let (X, t) be a T_c space. Let A be an g-closed set of (X, t). Then A is also gs-closed set. Since (X, t) is T_c, then A is a g*-closed set of (X, t). Therefore (X, t) is an _aT_c space.

..........The converse of the above theorem is not true as can be seen from the following example.

..........EXAMPLE 4.39 - Let X = {a, b, c} and t = {f, X, {a}, {b},{a, b}}. (X, t) is not a T_c space since {b} a gs-closed set but not a g*-closed set of (X, t). However (X, t) is an _aT_c space.

..........PROPOSITION 4.40 - If (X, t) is a T_c space, then for each x Î X, {x} is semi-closed or g*-open in (X, t).

PROOF: Let (X, t) be a T_c space and suppose that {x} is not semi-closed for some x Î X. Then by the proposition 6.4(ii) of [8], X-{x} is a sg-closed set. By (2.5) of [8], X-{x} is gs-closed. Since (X, t) is a T_c space, then X-{x} is g*-closed or equivalently, {x} is g*-open.

..........REMARK 4.41 - The following example shows that the converse of the above Proposition is not true in general.

..........EXAMPLE 4.42 - Let X and t be as in the example 4.39. Each singleton {x} of X is semi-closed. Already we have seen that (X, t) is not a T_c space.

..........Next we prove that the dual of the class of T_d spaces to the class of T_c spaces is the class of _aT_c spaces.

..........THEOREM 4.43 - A space (X, t) is T_c if and only if it is a T_d and an _aT_c.

PROOF: Necessity - Follows from the theorem 4.37 and 4.38.

Sufficiency - Suppose (X, t) is both T_d and _aT_c. Let A be a gs-closed set of (X, t). Since (X, t) is T_d, then A is g-closed. Then A is also an ag-closed set of (X, t). Since (X, t) is an _aT_c, then A is a g*-closed set of (X, t). Therefore (X, t) is a T_cspace.The following theorem shows that the dual of the class of T_c spaces to the class of T_b spaces is the class of T_1/2* spaces.

..........THEOREM 4.44 - A space (X, t) is a T_b space if and only if it is a T_c and a T_1/2* space.

PROOF: Necessity - Follows from the theorem 4.13 and 4.36.

Sufficiency - Suppose (X, t) is both T_c and T_1/2*. Let A be a gs-closed set of (X, t). Since (X, t) is a T_c space, then A is a g*-closed set. Since (X, t) is a T_1/2* space, then A is a closed set of (X, t). Therefore (X, t) is a T_b space.

..........By considering three classes, namely the class of _aT_c spaces, the class of T_d spaces and the class of T_1/2* spaces, in the next theorem we have that the intersection of any two of these three classes is the dual of the other class to the class of T_b spaces.

..........THEOREM 4.45 - A space (X, t) is a T_b space if and only if it is _aT_c, T_d and T_1/2*.

PROOF: Follows from the theorems 4.43 and 4.44.

4.46

[see my original paper for this diagram]

..........THEOREM 4.47 - For a topological space (X, t), the following conditions are equivalent.

(1) (X, t) is a T_1/2 space.

(2) Every singleton of X is either open or closed.

(3) Every singleton of X is either semi-open or closed.

(4) Every ag-closed set of (X, t) is a-closed.

(5) Every gs-closed set of (X, t) is semi-closed.

(6) Every q-generalized closed set of (X, t) is closed.

(7) (X, t) is a semi-T1/2 space and a semi-pre-T1/2 space.

(8) (X, t) is a T_1/2* space and *T_1/2 space.

PROOF: (1) Û (2) is nothing but the theorem 2.5 of [14]. From the theorem 6.5 of [8], we have (1) Û (3) Û (5). By the theorem 2.3 of [10], we have (1) Û (4). (1) Û (7) is nothing but the theorem 4.5 of [11].By the theorem 4.30, we have (1) Û (8). (1) Û (6) follows from [13].

§ 5. g*-CONTINUOUS AND g*-IRRESOLUTE MAPS

..........We introduce the following definition.

..........DEFINITION 5.01 - A function f :(X, t) ® (Y, s) is called g*-continuous if f ^-1(V) is a g*-closed set of (X, t) for every closed set V of (Y, s).

..........THEOREM 5.02 - Every continuous map is g*-continuous.

..........The following example supports that the converse of the above theorem is not true in general.

..........EXAMPLE 5.03 - Let X = Y = {a, b, c}, t = {f, X, {a}, {a, c}} and s = {f, X, {a}, {b}, {a, b}}. Define f :(X, t) ® (Y, s) by f(a) = a, f(b) = c and f(c) = b. f is not continuous since {a, c} is a closed set of (Y, s) but f ^-1({a, c}) = {a, b} is not a closed set of (X, t). However f is g*-continuous.

..........Thus the class of all g*-continuous maps properly contains the class of all continuous maps. Next we will observe that the class of all g*-continuous maps is properly contained in the classes of g-continuous maps, ag-continuous maps, a**g-continuous maps, gs-continuous maps, gsp-continuous maps, gp-continuous maps, r-g-continuous maps and gpr-continuous maps.

..........THEOREM 5.04 - Every g*-continuous map is g-continuous and hence an ag-continuous, a**g-continuous, gs-continuous, gsp-continuous, gp-continuous, r-g-continuous and gpr-continuous.

PROOF: Let f : (X, t) ® (Y, s) be a g*-continuous map. Let V be a closed set of (Y, s). Since f is g*-continuous, then f ^-1(V) is a g*-closed set of (X, t). By the theorem 3.04 and 3.06, f ^-1(V) is g-closed, ag-closed, a**g-closed, gs-closed, gsp-closed, gp-closed, r-g-closed and gpr-closed set of (X, t).

..........The following example supports that the converse of the above theorem is not ture in general.

..........EXAMPLE 5.05 - Let X = {a, b, c} and t = {f, X, {a}, {b, c}}. Define g :(X,t) ® (X, t) by g(a) = b, g(b) = a and g(c) = c. GC(X, t) = P(X, t) and G*C(X, t) =C(X, t). g is g-continuous and hence ag-continuous, a**g-continuous gs-continuous, gsp-continuous, gp-continuous, r-g-continuous and gpr-continuous. But g is not g*-continuous.

..........REMARK 5.06 - g*-continuity and b-continuity are independent as the next two examples show.

..........EXAMPLE 5.07 - Let X, Y, t, s and f be as in the example 5.03. Then C(Y, s) = {f, Y, {c}, {a, c},{b, c}}. G*C(X, t) = {f, X, {b}, {a, b},{b, c}}. f is not b-continuous since {a, c} is a closed set of (Y, s) but f ^-1 ({a, c}) = {a, b} is not a b-closed set of (X, t). However f is g*-continuous.

..........EXAMPLE 5.08 - Let X = Y = {a, b, c}, t = {f, X, {b}, {a, c}} and s = {f, Y, {a}, {a, b}, {a, c}}. Let i : (X, t) ® (Y, s) be the identity map. C(Y, s) = {f, Y, {b}, {c}, {b, c}} = SPC(X, t). i is not a g*-continuous map since {c} is a closed set of (Y, s) but i ^-1({c}) = {c} is not a g*-closed set of (X, t). However i is b-continuous.

..........THEOREM 5.09 - g*-continuity is independent from semi-continuity, a-continuity and pre-continuity.

PROOF: The function f in the example 5.07 is g*-continuous. f is neither semi-continuous nor pre-continuous. So f is not an a-continuous map. The function i in the example 5.08 is a-continuous and hence it is both semi-continuous and pre-continuous. But already we observed that i is not g*-continuous.The following example supports that the composition of two g*-continuous map need not be g*-continuous map again.

..........EXAMPLE 5.10 - Let X = Y = Z = {a, b, c}, t = {f, X, {a}, {b}, {a, b}, {a, c}}, s = {f, Y, {a}, {a, c}} and h = {f, X, {a}, {b}, {a, b}}. Let f be the identity map from (X, t) onto (Y, s). Define g :(Y, s) ® (Z, h) by g(a) = a, g(b) = c and g(c) = b . Clearly f and g are g*-continuous. {a, c} is a closed set of (Z, h) but (f o g) ^-1 ({a, c}) = f ^-1(g ^-1({a, c})) = f ^-1 ({a, b}) = {a, b} is not a g*-closed set of (X, t). Therefore f o g : (X, t) ® (Z, h) is not g*-continuous.

..........DEFINITION 5.11 - A function f : (X, t) ® (Y, s) is called g*-irresolute if f ^-1(V) is a g*-closed set of (X, t) for every g*-closed set of (Y, s).

..........THEOREM 5.12 - Every g*-irresolute function is g*-continuous.

..........The following example supports that the converse of the above theorem is not true.

..........EXAMPLE 5.13 - Let X = Y = {a, b, c}, t = {f, X, {a}, {b}, {a, b}, {a, c}} and s = {f, Y, {a, b}}. Define h : (X, t) ® (Y, s) by h(a) = b, h(b) = c and h(c) = a. h is not a g*-irresolute map since {b, c} is a g*-closed set of (Y, s) but h^-1({b, c}) = {a, b} is not a g*-closed set of (X, t). However h is g*-continuous.

..........THEOREM 5.14 - Let f : (X, t) ® (Y, s) and g :(Y, s) ® (Z, h) be any two functions. Then

(1) g o f is g*-continuous if g is continuous and f is g*-continuous.

(2) g o f is g*-irresolute if both f and g are g*-irresolute.

(3) g o f is g*-continuous if g is g*-continuous and f is g*-irresolute.

..........THEOREM 5.15 - Let f : (X, t) ® (Y, s) be a g*-continuous map. If (X, t) is T_1/2*, then f is continuous .

..........THEOREM 5.16 - Let f : (X, t) ® (Y, s) be an ag-continuous map. If (X, t) is _aT_c, then f is g*-continuous.

..........THEOREM 5.17 - Let f : (X, t) ® (Y, s) be a g-continuous map. If (X, t) is *T_1/2, then f is g*-continuous.

..........THEOREM 5.18 - Let f : (X, t) ® (Y, s) be a gs-continuous map. If (X, t) is T_c, then f is g*-continuous.

..........THEOREM 5.19 - Let f : (X, t) ® (Y, s) be a gc-irresolute and closed map. Then f(A) is a g*-closed set of (Y, s) for every g*-closed set A of (X, t).

PROOF: Let A be a g*-closed set of (X, t), Let U be a g-open set of (Y, s) such that f(A) Í U. Since f is gc-irresolute, then f ^-1(U) is a g-open set of (X, t). Since A Í f ^-1(U) and A is a g*-closed set of (X, t), then cl(A) Í f ^-1(U).Then f(cl(A)) Í f (f ^-1(U)) Í U. Since f is closed, then f(cl(A) = cl(f(cl(A)). This implies cl(f(A) Í cl(f(cl(A))) Í f(cl(A) Í U. Therefore f(A) is a g*-closed set of (Y, s).

..........THEOREM 5.20 - Let f : (X, t) ® (Y, s) be onto, g*-irresolute and closed. If (X, t) is T_1/2*, then (Y, s) is also a T_1/2* space.

..........DEFINITION 5.21 - A function f : (X, t) ® (Y, s) is called a pre-g*-closed if f(A) is a g*-closed set of (Y, s) for every g*-closed set of (X, t).

..........THEOREM 5.22 - Let f : (X, t) ® (Y, s) be onto, gc-irresolute and pre-g*-closed. If (X, t) is *T_1/2, then (Y, s) is also a *T_1/2 space.

..........THEOREM 5.23 - Let f : (X, t) ® (Y, s) be onto, ag-irresolute and pre-g*-closed. If (X, t) is an _aT_c space, then (Y, s) is also an _aT_c space.

..........THEOREM 5.24 - Let f : (X, t) ® (Y, s) be onto, gs-irresolute and pre-g*-closed. If (X, t) is a T_c space, then (Y, s) is also a T_c space.

Acknowledgement:

.......I am thankful to Professor J.Umehara for his valuable comments and suggestions. I am also thankful to Professors J.Dontchev(Finland), H.Maki(Japan) and M.Ganster(Austria) for sending many of their reprints as soon as I requested.

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