Open Journal of Discrete Mathematics, 2012, 2, 142144 http://dx.doi.org/10.4236/ojdm.2012.24028 Published Online October 2012 (http://www.SciRP.org/journal/ojdm) A Note on WeaklyαIFunctions and WeaklyαIParacompact Spaces* ShiQin Liu Department Mathematics and Computer, Hengshui College, Hebei, China Email: liushiqin168@163.com Received August 9, 2012; revised September 3, 2012; accepted September 10, 2012 ABSTRACT This paper introduces the new notion of weaklyαIfunctions and weaklyαIparacompact spaces in the ideal topo logical space. And it obtains that some properties of them. Keywords: WeaklyαIOpen Set; WeaklyαIFunctions; WeaklyαIParacompact Spaces 1. Introduction Throughout this paper, ClA and ntA denote the closure and interior of , respectively. Let ,X be a topological space and let be an ideal of subsets of . An ideal topological space is a topological space X , with an ideal on , and is denoted by ,, I . For a subset X, for *each neighborhood of IxXU x AI is called the local function of with respect to and [1]. It is well known that defines a Kuratowski closure operator for ** =AA A Cl . Let be a subset of a topological space S ,X . The complement of a semiopen set is said to be semi closed [2]. The intersection of all semiclosed sets con taining , denoted by S Cl S is called the semiclo sure [3] of . The semiinterior of , denoted by S S Int S, is defined by the union of all semiopen sets contained in . S In recent years, E. Hatir and T. Noiri have extended the study to αIopen and semiIopen sets. In this paper, we introduce the new sets which are called weakly αIopen and weaklyαIfunctions, then obtain some properties of t h em. First we recall some definitions used in the sequel. Definition 1.1. [4] A subset of an ideal topological space ,, I is said to be weaklyαIopen, if AsClIntClIntA . Definition 1.2. [4] A subset of an ideal topological space S ,, I is said to be a weaklyαIclosed set if its complement is a weaklyαIopen set. Theorem 1.1. [4] Let ,, I be an ideal top ological space. Then all weaklyαIopen sets constitute a topo logy of . Then (1) and are weaklyαIopen sets. (2) The finite intersection of weaklyαIopen sets are weaklyαIopen sets. (3) If is weaklyαIopen for each , then is weaklyαIopen. Theorem 1.2. [4] Let ,, I be an ideal topological space and AU . Then is weaklyαIopen if and only if is weaklyαIopen in ,, UU UI . 2. WeaklyαIFunctions Definition 2.1. A function :,, ,fXI Y 1 is said to be weaklyαIcontinuous , if V is weakly αIopen in ,, I , for any ,VY . Definition 2.2. A mapping ,,:,, XIYJ U is said weaklyαIopen (resp weaklyαIclosed) if for any U is weaklyαIopen (resp. is weaklyαIclosed) U is weaklyαIopen (resp. U is weaklyαI closed). Theorem 2.1. For a function :,, ,fXI Y , the followings are equivalent: (1) is weaklyαIcontinuous. (2) For any X and each V containing x, there exists weaklyαIopen containing U such that UV. (3) The inverse image of each closed set in is weakly αIclosed. Proof: 13 it is obviously. 12 For each X and each weaklyαI open ,VY containing x. Since is weakly αIcontinuous, then 1 V is a weaklyαIopen set containing in . Let *Supported by Hebei Province of the Scientific Research in mentoring rograms Z2010187. 1 =UfV , then UV. C opyright © 2012 SciRes. OJDM
S.Q. LIU 143 21 For any X and each V contain ing x, there exists weaklyαIopen U containing such that x U 1 V and . 1V x Uf We have 1 xf V V U , where U is weakly αIopen for any 1 fG. Thus is weakly αIopen by Theorem 1.1. And conclude that is weakly αIcontinuous. fV f 1 Theorem 2.2. Let :,, ,fXI Y be a fun ction and U an open cover of . Then is weaklyαIcontinuous if and only if the restriction f :,, , UUU fU IY is weaklyαIcontinuous for each . Proof: Necessity. Let be any open set of V ,Y . Since is weaklyαIcontinuous, 1 V is a weakly αIopen set of ,, I . Since U , from Theorem 1.2 is weaklyαIopen in 1 UfV ,, UU UI . On the other hand, 11 U VU fV and 1 U V is weakly αIopen in ,, UU UI . This shows that U is weaklyαIcontinuous for each . Sufficiency. Let be any open set of V ,Y . Since U is weaklyαIcontinuous for each . 1 U V is weaklyαI open of ,, UU UI and hence by Theorem 1.2. 1 U V is weakly open in ,, I for each . Moreover, we have 11 1 U 1 VUfVUfV fV . Therefore 1 V is a weakly open in ,, I by Theorem 1.1. This shows that is weaklyαIconti nuous. Theorem 2.3. A function is weaklyαIcontinuous if and only if the graph function : XXY defined by , xxfx for each X, is weaklyαIcontinuous. Proof: Necessity. Suppose that is weaklyαI con tinuous. Let X and W be any open set of Y containing x. Then there exists a basic open set such that UV , xxf UVWx. Since f is weaklyαIcontinuous, there exists a weaklyαIopen set 0 of U containing such that 0 U U V. By Theorem 1.1. is weaklyαIopen and 0 U 0 UU UV W. This show that is weakly αIcontinuous. Sufficiency. Suppose that is weaklyαIcontinuous. Let X and V be any open set of containing Y x. Then V is open in Y and by the weaklyαIcontinuous of ,there exists a weaklyαI open set U containing such that UXV. Therefore we obtain UV. This shows that is weaklyαIcontinuous Theorem 2.4. Let :,, ,, XIY J is weakly αIopen (resp weaklyαIclosed) mapping. If Y and is a weaklyαIclosed (resp weaklyαIopen) set of U containing 1 y V , then ther e exists a w ea kl y αIclosed (resp weaklyαIopen) subset of con taining such that Y y 1 VU . Proof: Suppose that is weaklyαI c l osed m a ppin g. Given Y and U is a weaklyαIopen subset of containing 1 y , then U is a weaklyαI closed set. Since is weaklyαIclosed, XU U is weaklyαIclosed. Hence VY is weakly αIopen. It follows fro m X 1f yU that UyfX . Therefore YfXU V and 11 1 VfYfXU XffXUU . Similar argument holds for a weaklyαIopen map ping. 3. WeaklyαIParacompact Spaces Definition 3.1. A space is said to be weaklyαI Hausdorff, if for each pair of distinct points and in y , there exist disjoint weaklyαIopen sets and in U V such that U and V, and UV . Definition 3.2. A space is said to be weaklyαI regular space, if for every X and every weaklyαI closed set X such that F , there exist weakly αIopen sets , such that 1 U2 U1 U, 2 U and 12 UU . Definition 3.3. A space is said to be weaklyαI normal space, if for every pair of disjoint weaklyαI closed sets , , there exist weaklyαIop e n s e t s , such that BX U V U, and UVBV . Definition 3.4. An ideal topological space ,, I is said to be a weaklyαIcompact space if every weakly αIopen cover of has a finite subcover. Definition 3.5. An ideal topological space ,, I is said to b e weaklyαIparacompact space, if every w eak  lyαIopen cover of has a locally finite weaklyαI open refinem ent . Definition 3.6. Mapping :,, ,, XIY J is said to be weaklyαIperfect, if is weaklyαIclosed and for any Y , 1 y is a weaklyαIcompact subset of . Theorem 3.1. An ideal topological space ,, I is Copyright © 2012 SciRes. OJDM
S.Q. LIU Copyright © 2012 SciRes. OJDM 144 s S U , and we conclude that is a weaklyαI a weaklyαIcompact space if and only if every family of weaklyαIclosed sets of X satisfying the finite in tersection property has nonempty intersection. compact space. Lemma 3.1. Let be a weaklyαIparacompact space and , a pair of weaklyαIclosed sets of B . If for every B there exist weaklyαIopen sets U, V such that U, V and xx UV U V . Then there also exist weaklyαIopen sets , such that U, and UV . BV Proof: Necessity. If is any family of weaklyαI closed sets which has finite intersection property, and F F , then FF FXF X. Thus F is a weaklyαIopen set. Since is a weaklyαIcompact space, hence there exist finite Proof: sets 12 ,,, n FF, such that 1 n i i FX . So 1 n i i FX , and , a contradiction. 1 n i i F Sufficiency. If is any weaklyαIopen cover for , then is a weaklyαIclosed family that satisfies :A XA FA A FXAXA . So dose not satisfy finite intersection property, which means has a finite subfamily 12 ,,, n FF ,1 ii which has intersection empty. Suppose ,2,, XAi nn n, where i A. We have . So 11 1 n iii ii i FXAXA ,,, 12 n AA is a finite cover of . Theorem 3.2. WeaklyαIcompactness is an inverse invariant of weaklyαIperfect mapping. Proof: Let : XY be a weaklyαIperfect map ping onto a weaklyαIcompact space Y. Given a weaklyαIopen cover s S U of the space X. Since is a weaklyαIperfect mapping and for every Y choose a finite set , such that y SS 1 y sS y yU U. U is a weaklyαIop en from Theorem 1.1. And from Theorem 2.4 there exists a weakly αIopen set V containing such that y 11 y yfV U . Since is a weaklyαIcom  pac t space, the weakly Y open cover y Y V of has a finite subcover such that Y 1,,k:= yi Vi 1 k yi i YV . Therefore 1 111 kkk yy ii iiisS yi s fV UU 1 k , which means isS yi U . Because the family of these () :,,, sy i UsS iik is a finite subfamily of x B XB V is a weaklyαIopen cover of the weaklyαIparacompact space , so that it has a locally finite weaklyαIopen refinement s S W . Let :, s SxBWV 1x Ss ,, then for any 1 S , s AW and 1 sS BW . Since s S W is a locally finite family, it is a closure preserving family. So 11 s sS sS WW , then the set 1 sS UXW is open. Therefore U is a weaklyαI open set. 1 sS VW UV is also a weaklyαIopen set from theorem1.1. and . Theorem 3.3. Assuming is a weaklyαIpara compact space, if onepoint sets of are weaklyαI closed sets, then is weaklyαInormal space. Proof: Substituting onepoint sets for A in Lemma 4. 1., and if onepoint sets are weaklyαIclosed sets, we see that every weaklyαIparacompact space is weaklyαI regular. Applying Lemma 3.1. again we have the con clusion. 4. Summary Combining the topological structure with other mathe matical features has provided many interesting topics in the development of general topology. And this paper has done much work on the ideal topological space. On certain extent, it promotes the development of topology. REFERENCES [1] D. Jankovic and T. R. Hamlet, “New Topological from Old via Ideals,” The American Mathematical Monthly, Vol. 97, No. 4, 1990, pp. 295310. doi:10.2307/2324512 [2] S. G. Grossley and S. K. Hidebrand, “SemiClosure,” Texas Journal of Science, Vol. 22, No. 23, 1971, pp. 99112. [3] E. Hatir, A. Keskin and T. Noiri, “A Note on βISets and Strongly βIContinuous Functions,” Acta Mathematica Hungarica, Vol. 108, No. 12, 2005, pp. 8794. doi:10.1007/s1047400502102 [4] Q. L. Shi, “On the WeaklyαI Open Sets,” Acta Mathe matica Hungarica.
