Abstract:

Introducing  and investigating  a new class of sets called Pp -open sets is the main aim of this work and using this set we define and study new classes of the most well- known notions  topological space such as continuity  and new types of separation axioms.

We start  this work by defining the class of Pp -open sets which is contained  in the class of preopen  sets and  contains  the  class of pre-θ-open sets.  It is shown that the family of Pp O(X ) sets form a topology on a space X , if X is locally indiscrete or has a property  P . It is also appeared  that  the family of Pp O(X ) sets is discrete topology in X , if (X, τ ) is indiscrete  topology.  It has also been proved that  the family of Pp -open sets  and  preopen  sets  are  identical,  if X  is pre-T1   or locally indiscrete  spaces.  It is shown that  if X  is pre-regular,  then  τ ⊆ Pp O(X ) and for any subset  A of a space X , we have Pp Bd [Pp Bd{Pp Bd(A)}] = Pp Bd [Pp Bd(A)]. In the  last  section,  the  Pp -g.closed sets  have  been defined and  characterization theorems  are provided in some spaces.

Moreover, some new types  of separation  axioms are provided  and  defined.  It  is

proved  that  Pp -T1  gives Pp -T 1 2 and  Pp -T 1 2 gives Pp -T0 .  It  is shown that  a space

X  is Pp -T 1 2 if and  only if every singleton  is either  Pp -closed or Pp -open.  It  has

also been proven that  the notions of Pp -D1  and Pp -D2  are equivalent.  In addition, the  properties  and  characterizations of Pp -R0 , Pp -R1   and  Pp -regular  spaces are studied.

Finally, the concepts of Pp -continuous and weakly Pp -continuous are defined. It is proved that  Pp -continuity  is weaker than  strongly θ-continuous, but  it is stronger than  precontinuity.  It  is observed that  weakly Pp -continuity  is weaker than  Pp – continuity  and stronger than  weakly precontinuity. Furthermore, many conditions are provided that  make Pp -continuity  and weakly Pp -continuity  equivalent to some other  types  of continuity.   It  is demonstrated that  the  inverse image of each D- set in Y  is a Pp D-set  in X , if f : X → Y  is a Pp -continuous  surgective function. Moreover, it is perceived that  X is a Pp -D1  space, if Y  is a D-set and f : X → Y  is a Pp -continuous bijective function.  It has been proven that  there are relationships between  X  and  Y   with  respect  to  separation  axioms,  if f : X  → Y   is a Pp – continuous.

Bibliography

[1] M. E. Abd El-Monsef, S. N. El-Deeb and R. A. Mahmoud,  β-open sets and β-continuous  mappings, Bull. Fac.  Sci. Assuit. Univ., 12 (1) (1983), 1-18.

[2] N. K. Ahmed, On some types of separation  axioms, M.Sc. Thesis, College of Science, Salahaddin  Univ., 1990.

[3] D. Andrijevic,  Some properties  of the  topology of α-sets,  Math.  Vesnik, 36 (1984), 1-10.

[4] C.  W.  Baker,  Weakly  θ-precontinuous  functions,  Acta  Math.  Hungar.  100 (2003), 342-351.

[5] M. Caldas,  S. Jafari  and  T.Noiri,  Characterizations of Pre-R0  and  Pre-R1 Topological Spaces, Topology Proceedings,  Vol. 25, Summer 2000, 17-30.

[6] S. H. Cho, A note on strongly θ-precontinuous functions, Acta Math. Hungar., 101 (1-2)(2003), 173-178.

[7] J. Dontchev, Survey on preopen sets, The Proceedings of the Yatsushiro Topo- logical Conference,  (1998), 1-18.

[8] S. N. El-Deeb,  I. A. Hasanein,  A. S. Mashhour  and  T.  Nori, On P-regular spaces, Bull. Math. Soc. Sci. Math. R.S. Roum, 27 (4) (1983), 311-315.

[9] S. Jafari,  On a weak separation  axiom, Far  East  J. Math. Sci., 3 (5) (2001), 779-787.

[10] J. E. Joseph and M. H. Kwack, On S-closed spaces, Proc.  Amer. Math. Soc., 80 (2) (1980), 341-348.

[11] A. Kar and P. Bhattacharyya, Some weak separation axioms, Bull. Cal. Math. Soc., 82 (1990), 415-422.

[12] G. Di Maio, On semi topological operator  and semi separation  axioms, Rend. Circ.  Math. Palermo  (2) Suppl. Second Topology Conference,  12 (1986), 219-230.

[13] N. Levine, A decomposition of continuity in topological space, Amer.  Math. Monthly 68 (1961), 44-46.

[14] N. Levine, Semi-open sets and semi-continuity  in topological spaces, Amer. Math. Monthly, 70 (1) (1963), 36-41.

[15] P.  E.  Long and  L. Herrington,  Strongly  θ-continuous  functions,  J. Korean Math. Soc., 18 (1) 1981, 21-28.

[16] H. Maki, J. Umehara  and T. Noiri, Every topological space is Pre-T 1 , Mem. 2 Fac.  Sci. Kochi Univ. Ser. A Math., 17 (1996), 33-42.

[17] A. S. Mashhour,  M. E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and week precontinuous  mappings, Proc.  Math. Phys. Soc. Egypt, 53 (1982), 47-53.

[18] A. A. Nasef and T. Noiri, Some weak forms of almost continuity,  Acta Math. Hungar.,  74 (3) (1997), 211-219.

[19] G. B. Navalagi, Pre-neighbourhoods, the Mathematics Education,  32 (4), Dec. (1998), 201-206.

[20] G. B. Navalagi, Further Properties  of Pre-T0, Pre-T1,and  Pre-T2   Spaces, In- ternational Journal  of Mathematics  and Computing Applications  Vol. 3, Nos. 1-2, January-December 2011, pp. 67-75.

[21] O. Njastad,  On  some classes of nearly  open sets,  Pacific  J. Math.,  15 (3) (1965), 961-970.

[22] T.  Noiri,  Strongly  θ-precontinuous  functions,  Acta  Math.  Hungar.,  90 (4) (2001), 307-316.

[23] J. H. Park  and Y. B. Park,  On sp-regular spaces, J. Indian  Acad Math., Vol. 17 (2) (1995), 213-218.

[24] M. Pal and P. Bhattacharyya, Feeble and strong forms of pre-irresolute  func- tion,Bull.  Malaysian  Math. Soc. (Second Series), 19 (1996), 63-75.

[25] R. Paul and P. Bhettecharyya, On pre-Urysohn spaces, Bull. Malaysian Math. Soc., (Second Series) 22(1999) 23-34.

[26] I. L. Reilly and M. K. Vamanmurthy, On α-continuity in topological spaces, Acta.  Math. Hungar.,  45 (1-2) (1985), 27-32.

[27] H. A. Shareef, SP -open sets, SP -continuity  and SP -compactness in topological spaces, M. Sc. Thesis,  Sulaimani  University 2007.

[28] M. H. Stone, Applications  of the theory  of boolean rings to topology, Trans. Amer. Math. Soc., 41 (1937), 375-481.

[29] J. Tong, A separation  axioms between T0 and T1, Ann. Soc. Sci. Bruxelles 96 (2) (1982), 85-90.

[30] N. V. Velicko, H-closed topological spaces, Amer.  Math. Soc. Transl., 78 (2) (1968), 103-118.

[31] R. H. Yunis,  Properties  of θ-semi-open sets,  Zanco  J. of Pure  and  Applied

Sciences, 19 (1) (2007), 116-122.

€59.00
Purchase