On ḡ -Closed Mappings in Topological Spaces

Download Full-Text PDF Cite this Publication

Text Only Version

 

On ḡ -Closed Mappings in Topological Spaces

Manoj Garg, Shailendra Singh Rathore

    1. Department and Research Centre of Mathematics, Nehru P. G. College, Chhibramau, Kannauj, U.P., India

      Abstract:- In this paper we introduce a new class of closed maps namely g -closed maps which settled in between the class of closed maps and the class of g-closed maps and then we study many basic properties of g -closed maps together with the relationships of some other maps.

      2000 Mathematics Subject Classification: 54c10, 54c20.

      Key words and phrases: g -closed maps, g *-closed maps

      1. INTRODUCTION

        Malghan(22) and Devi et al(8) introduced the concept of generalized closed maps and semi generalized closed maps respectively in topological spaces. Manoj et al(23) introduced the concept of g -closed sets in topological spaces. In this paper

        we introduce a new class of closed maps namely g -closed maps and g *-closed maps.

      2. PRELIMINARIES

        Throughout this paper (X, ), (Y, ) and (Z, ) represent topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of space (X, the cl(A), int(A) and Ac denote the closure of A, the interior of A and the complement of A in X respectively.

        We recall the following definitions:

        Definition 2.01: A subset A of a topological space (X, is called g-closed(2) (resp. *g-closed(13), g*-closed(13), **g-closed(17), g -closed(12)) set if cl(A) U (resp. cl(A) U, cl(A) U, scl(A) U, cl(A) U) whenever A U and U is open (resp. g – open, g-open, g -open, sg-open) set in (X,

        Definition 2.02 : A map f : (X, (Y, ) is called g-closed(20) (resp. *g-closed(13), g*-closed(24), **gs-closed(8), g -closed(12)) map if the image of each closed set in (X, ) is g-closed (resp. *g-closed, g*-closed, **gs-closed, g -closed) in (Y, ).

      3. g -CLOSED MAPS In this section we introduce the following definitions.

Definition 3.01: A map f : (X, (Y, ) is called g -closed (resp. g -open) map if f(A) is g -closed (resp. g -open) set in (Y,

) for every closed (open) set A of (X, .

Definition 3.02: Let (X, ) be a topological space and A X. We define the g -interior of A (briefly g -int(A)) to be the union of all g -open sets contained in A.

Theorem 3.03: Every closed map, *g-closed map, g*-closed map and g -closed map is g -closed map.

Next examples show that the converse of the above theorem is not true in general.

Example 3.04: Let X = Y = {a, b, c}, , {c}, {a, c}, {b, c}, X} and , {a}, {b, c}, Y}. Define f : (X, (Y, ) by f(a) = b, f(b) = a and f(c) = c, then f is not closed map, *g-closed map, g*-closed map and g -closed map however f is g -closed map.

Theorem 3.05: Every g -closed map is g-closed map and **gs-closed map.

Example 3.06: Let X = Y = {a, b, c}, , {a}, {a, b}, {a, c}, X} and , {a}, {a, b}, Y}. Define f : (X, (Y, ) by identity mapping, then f is not g-closed map and **gs-closed map however f is g -closed map.

Therefore the class of g -closed maps properly contains the class of closed maps, the class of *g-closed maps, the class of

g -closed maps and the class of g*-closed maps and properly contained in class of g-closed maps and the class of **gs-closed maps.

Theorem 3.07: If f : (X, (Y, ) be a closed map and g : (Y, (Z, ) be a g -closed map then their composition gof : (X, (Z, ) is g -closed map.

Remark 3.08: The following example shows that the composition of two g -closed maps need not be g -closed map.

Example 3.09: Let X = {a, b, c}, , {a}, {b, c}, X}, , {b}, X} and , {a}, {b}, {a, b}, X}. Define f : (X, (X, ) by f(a) = b, f(b) = c and f(c) = a. Define g : (X, (X, ) by identity mapping then f and g both are g -closed maps but their composition gof : (X, (X, ) is not a g -closed map.

Theorem 3.10: If f : (X, (Y, ) and g : (Y, (Z, ) be two mappings such that their composition gof : (X, (Z, ) be a g -closed map then the following are true

  1. If f is continuous and surjective, then g is g -closed map.
  2. If g is g -irresolute and injective, then f is g -closed map.

Theorem 3.11: For any bijective f : (X, (Y, ) the following statements are equivalent.

  1. f-1 : (Y (X, ) is g -continuous.
  2. f is g -open map and
  3. f is g -closed map.

Definition 3.12: A map f : (X, (Y, ) is said to be a g *-closed (resp. g *-open) if the image f(A) is g -closed (resp. g – open) set in (Y, ) for every g -closed (resp. g -open) set A in (X, ).

Theorem 3.13: Every g *-closed map is g -closed map.

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

Example 3.14: Let X = Y = {a, b, c}, , {a}, {b, c}, X} and , {b}, Y}. Define f : (X, (Y, ) by f(a) = b, f(b) = c and f(c) = a then f is g -closed map but not g *-closed map.

Theorem 3.15: For any bijection f : (X (Y, ) the following are equivalent

  1. f-1 : (Y, (X, ) is g -irresolute,
  2. f is a g *-open map and
  3. f is a g *-closed map.

REFERENCES

  1. Levine N.: Semi open sets and semi continuity in topological spaces, Amer. Math. Monthly, 70 (1963), 36-41.
  2. Levine N: Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19 (1970), 89-96.
  3. Andrijevic D.: Semi-preopen sets Mat. Vesnik, 38(1) (1986) 24-32.
  4. Bhattacharya P. and Lahiri B.K.: Semi generalized closed sets in a topology, Indian J. Math. 29 (3), (1987) 375-382.
  5. Arya S.P. and Tour N.: Characterizations of s-normal spaces, Indian J. Pure Applied Math., 21(8) (1990), 717-719.
  6. Balachandran K., Sundaram P. and Maki H.: On generalized continuous maps in topological space, Mem. Fac. Sci. Kochi Univ. Ser. A. Math. 12(1991), 5-13.
  7. Sundaram P., Maki H. and Balachandran K.: Semi-generalized continuous maps and semi-T1/2 spaces Bull. Fukukoa Univ. Ed. Part-III, 40(1991), 33- 40.
  8. Devi R., Maki H and Balachandran K.: Semi generalized closed maps and generalized semi closed maps Mem. Fac. Sci. Kochi Univ. Ser. A. Math. 14, (1993), 41-54.
  9. Dontchev. J.: On generalizing semi-preopen sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Math. 16(1995), 35-38.
  10. Devi R., Maki H and Balachandran K.: Semi generalized homeomorphism and generalized semi homeomorphism in topological spaces, Indian J. Pure Applied. Mat. 26, (3) (1995), 271-284.
  11. Veera Kumar M.K.R.S.: Between semi closed sets and semi pre-closed sets, Rend. Instint. Math. Univ. Trieste (Italy), XXXII(2000), 25-41.
  12. Garg M., Agarwal S. and Goel C.K.: On g -closed sets in topological spaces, Acta Ciencia Indica, XXXIII(M) (4) (2007), 1643-1652.
  13. Veera Kumar M.K.R.S.: Between g*-closed sets and g-closed sets, Ant.j.Math.(Reprints).
  14. Garg M., Agarwal S., Goel C.K. and Goel S., On g -homeomorphism in topological spaces, Ultra Sci.Phy.Sci.19(3)M(2007), 697-706.

(15) Biswas N.: Bull. Cal. Math. Soci., 61 (1969), 127-350.

  1. Garg M., Khare S. K. and S. Agarwal: On **g-closed sets in topological spaces, Ultra Sci.Phy.Sci. 20(2)M(2008), 403-416.
  2. Agarwal S., Garg M. and Goel C.K.: Between semi-closed sets and generalized semi closed sets in topological spaces, Ultra Sci.Phy.Sci.

    22(2)M(2010), 539-550.

  3. Agarwa S., Garg M. and Goel C.K.: On *gs-homeomorphisms in topological spaces, Jourl. Int. Acd. Phy. Sci., 11(2007), 63-70.
  4. Garg M., Agarwal S. and Goel C.K.: On -homeomorphisms in topological spaces, Reflection De Era , JPS, 4(2010), 9-24.
  5. Noiri T.: Atti. Accad. Zaz. Lincei. Rend. Cl. Sci. Fis. Mat. Nature, 54(8)(1973), 412-415.
  6. Goel C.K., Garg M. and Poonam Agarwal.: On **gs-closed sets in topological spaces, Ultra Sci. Phy. Sci., 27(2)B(2015), 95-104.
  7. Malghan S.R. : Generalized closed maps. J. Karnataka Univ. Sci. 27(1982), 82-88.
  8. Rathore S.S., Garg M. : A generalized closed sets in topological spaces, Jour. Current Adv. Res., 6(12)(2017), 8449-8453.
  9. Veera Kumar M.K.R.S.: Between closed sets and g-closed sets, Mam. Fac. Sci. Kochi Uni., Japan Sar. Math. 20(2000), 1-19.

Leave a Reply

Your email address will not be published. Required fields are marked *