 Open Journal of Discrete Mathematics, 2012, 2, 142-144 http://dx.doi.org/10.4236/ojdm.2012.24028 Published Online October 2012 (http://www.SciRP.org/journal/ojdm) A Note on Weakly-α-I-Functions and Weakly-α-I-Paracompact Spaces* Shi-Qin 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 weakly-α-I-functions and weakly-α-I-paracompact spaces in the ideal topo- logical space. And it obtains that some properties of them. Keywords: Weakly-α-I-Open Set; Weakly-α-I-Functions; Weakly-α-I-Paracompact Spaces 1. Introduction Throughout this paper, ClA and IntA denote the closure and interior of A, respectively. Let ,X be a topological space and let I be an ideal of subsets of X. An ideal topological space is a topological space X, with an ideal I on X, and is denoted by ,,XI. For a subset AX, for*each neighborhood ofAIxXU x AI is called the local function of A with respect to I and  . It is well known that defines a Kuratowski closure operator for **=AA AClI. Let be a subset of a topological space S,X. The complement of a semi-open set is said to be semi- closed . The intersection of all semi-closed sets con- taining , denoted by SsCl S is called the semi-clo- sure  of . The semi-interior of , denoted by S SsInt S, is defined by the union of all semi-open sets contained in . SIn recent years, E. Hatir and T. Noiri have extended the study to α-I-open and semi-I-open sets. In this paper, we introduce the new sets which are called weakly- α-I-open and weakly-α-I-functions, then obtain some properties of t h em. First we recall some definitions used in the sequel. Definition 1.1.  A subset A of an ideal topological space ,,XI is said to be weakly-α-I-open, if AsClIntClIntA. Definition 1.2.  A subset of an ideal topological space S,,XI is said to be a weakly-α-I-closed set if its complement is a weakly-α-I-open set. Theorem 1.1.  Let ,,XI be an ideal top ological space. Then all weakly-α-I-open sets constitute a topo- logy of X. Then (1)  and X are weakly-α-I-open sets. (2) The finite intersection of weakly-α-I-open sets are weakly-α-I-open sets. (3) If A is weakly-α-I-open for each , then A is weakly-α-I-open. Theorem 1.2.  Let ,,XI be an ideal topological space and AU. Then A is weakly-α-I-open if and only if A is weakly-α-I-open in ,,UUUI. 2. Weakly-α-I-Functions Definition 2.1. A function :,, ,fXI Y1 is said to be weakly-α-I-continuous , if fV is weakly- α-I-open in ,,XI , for any ,VY. Definition 2.2. A mapping ,,:,,fXIYJU is said weakly-α-I-open (resp weakly-α-I-closed) if for any U is weakly-α-I-open (resp. is weakly-α-I-closed) fU is weakly-α-I-open (resp. fU is weakly-α-I- closed). Theorem 2.1. For a function :,, ,fXI Y, the followings are equivalent: (1) f is weakly-α-I-continuous. (2) For any xX and each V containing fx, there exists weakly-α-I-open containing Ux such that fUV. (3) The inverse image of each closed set in is weakly- α-I-closed. Proof: 13 it is obviously. 12 For each xX and each weakly-α-I- -open ,VY containing fx. Since f is weakly- α-I-continuous, then 1fV is a weakly-α-I-open set containing x in X. Let *Supported by Hebei Province of the Scientific Research in mentoring programs Z2010187. 1=UfV, then fUV. Copyright © 2012 SciRes. OJDM S.-Q. LIU 143 21 For any xX and each V contain- ing fx, there exists weakly-α-I-open xU containing x such that xfU1V and . 1VxUfWe have 1xxf VfVU, where xU is weakly- α-I-open for any 1xfG. Thus is weakly- α-I-open by Theorem 1.1. And conclude that is weakly- α-I-continuous. fVf1Theorem 2.2. Let :,, ,fXI Y be a fun- ction and U an open cover of X. Then is weakly-α-I-continuous if and only if the restriction f:,, ,UUUfU IY is weakly-α-I-continuous for each . Proof: Necessity. Let be any open set of V,Y. Since f is weakly-α-I-continuous, 1fV is a weakly- α-I-open set of ,,XI. Since U, from Theorem 1.2 is weakly-α-I-open in 1UfV,,UUUI. On the other hand,  11UfVU fVand 1UfV is weakly- α-I-open in ,,UUUI. This shows that Uf is weakly-α-I-continuous for each . Sufficiency. Let be any open set of V,Y. Since Uf is weakly-α-I-continuous for each . 1UfV is weakly-α-I- open of ,,UUUI and hence by Theorem 1.2. 1UfV is weakly open in ,,XI for each . Moreover, we have  111U1fVUfVUfVfV . Therefore 1fV is a weakly open in ,,XI by Theorem 1.1. This shows that f is weakly-α-I-conti- nuous. Theorem 2.3. A function f is weakly-α-I-continuous if and only if the graph function :gXXY defined by ,gxxfx for each xX, is weakly-α-I-continuous. Proof: Necessity. Suppose that f is weakly-α-I- con- tinuous. Let xX and W be any open set of XY containing gx. Then there exists a basic open set such that UV,gxxf UVWx. Since f is weakly-α-I-continuous, there exists a weakly-α-I-open set 0 of UX containing x such that 0fUUV. By Theorem 1.1. is weakly-α-I-open and 0U0gUU UV W. This show that g is weakly- α-I-continuous. Sufficiency. Suppose that g is weakly-α-I-continuous. Let xX and V be any open set of containing Yfx. Then XV is open in XY and by the weaklyα-I-continuous of g,there exists a weakly-α-I- open set U containing x such that gUXV. Therefore we obtain fUV. This shows that f is weakly-α-I-continuous Theorem 2.4. Let :,, ,,fXIY J is weakly- α-I-open (resp weakly-α-I-closed) mapping. If yY and is a weakly-α-I-closed (resp weakly-α-I-open) set of UX containing 1fyV, then ther e exists a w ea kl y- α-I-closed (resp weakly-α-I-open) subset of con- taining such that Yy1fVU. Proof: Suppose that f is weakly-α-I- c l osed m a ppin g. Given yY and U is a weakly-α-I-open subset of X containing 1fy, then XU is a weakly-α-I- closed set. Since f is weakly-α-I-closed, fXUU is weakly-α-I-closed. Hence VY is weakly- α-I-open. It follows fro m X1ffyU that UyfX. Therefore yYfXU V and 11 1fVfYfXU XffXUU  . Similar argument holds for a weakly-α-I-open map- ping. 3. Weakly-α-I-Paracompact Spaces Definition 3.1. A space X is said to be weakly-α-I- Hausdorff, if for each pair of distinct points x and in yX, there exist disjoint weakly-α-I-open sets and in UVX such that xU and yV, and UV. Definition 3.2. A space X is said to be weakly-α-I- regular space, if for every xX and every weakly-α-I- closed set FX such that xF, there exist weakly- α-I-open sets , such that 1U2U1xU, 2FU and 12UU. Definition 3.3. A space X is said to be weakly-α-I- normal space, if for every pair of disjoint weakly-α-I- closed sets A, , there exist weakly-α-I-op e n s e t s , such that BXU VAU, and UVBV. Definition 3.4. An ideal topological space ,,XI is said to be a weakly-α-I-compact space if every weakly- α-I-open cover of X has a finite subcover. Definition 3.5. An ideal topological space ,,XI is said to b e weakly-α-I-paracompact space, if every w eak - ly-α-I-open cover of X has a locally finite weakly-α-I- open refinem ent . Definition 3.6. Mapping :,, ,,fXIY J is said to be weakly-α-I-perfect, if f is weakly-α-I-closed and for any yY, 1fy is a weakly-α-I-compact subset of X. Theorem 3.1. An ideal topological space ,,XI is Copyright © 2012 SciRes. OJDM S.-Q. LIU Copyright © 2012 SciRes. OJDM 144 ssSU, and we conclude that X is a weakly-α-I- a weakly-α-I-compact space if and only if every family of weakly-α-I-closed sets of X satisfying the finite in- tersection property has nonempty intersection. compact space. Lemma 3.1. Let X be a weakly-α-I-paracompact space and A, a pair of weakly-α-I-closed sets of BX. If for every xB there exist weakly-α-I-open sets xU, xV such that xAU, xxV and xxUVU V. Then there also exist weakly-α-I-open sets , such that AU, and UV . BVProof: Necessity. If is any family of weakly-α-I- closed sets which has finite intersection property, and FF, then FFXFXF X. Thus XF is a weakly-α-I-open set. Since X is a weakly-α-I-compact space, hence there exist finite Proof: sets 12,,,nFFF, such that 1niiXFX. So 1niiXFX, and , a contradiction. 1niiFSufficiency. If is any weakly-α-I-open cover for X, then is a weakly-α-I-closed family that satisfies :AXAFA AFXAXA   . So dose not satisfy finite intersection property, which means has a finite subfamily 12,,,nFFF,1ii which has intersection empty. Suppose ,2,,FXAinnn, where iA. We have . So 11 1niiiii iFXAXA  ,,,12 nAAA is a finite cover of . Theorem 3.2. Weakly-α-I-compactness is an inverse invariant of weakly-α-I-perfect mapping. Proof: Let :fXY be a weakly-α-I-perfect map- ping onto a weakly-α-I-compact space Y. Given a weakly-α-I-open cover ssSU of the space X. Since f is a weakly-α-I-perfect mapping and for every yY choose a finite set , such that ySS1sysSyfyUU. yU is a weakly-α-I-op en from Theorem 1.1. And from Theorem 2.4 there exists a weakly- α-I-open set yV containing such that y11yyfyfV U. Since is a weakly-α-I-com - pac t space, the weakly-YIopen cover yyYV of has a finite subcover such that Y1,,k:=yiVi1kyiiYV. Therefore 1111kkkyyiiiiisSyisXfV UU   1k, which means sisSyiXU . Because the family of these ():,,,syiUsS iik is a finite subfamily of xxBXB V is a weakly-α-I-open cover of the weakly-α-I-paracompact space X, so that it has a locally finite weakly-α-I-open refinement ssSW. Let :,sSxBWV1xSs ,, then for any 1sS, sAW and 1ssSBW. Since ssSW is a locally finite family, it is a closure preserving family. So 11sssS sSWW, then the set 1ssSUXW is open. Therefore U is a weakly-α-I- open set. 1ssSVWUV is also a weakly-α-I-open set from theorem1.1. and . Theorem 3.3. Assuming X is a weakly-α-I-para- compact space, if one-point sets of X are weakly-α-I- closed sets, then X is weakly-α-I-normal space. Proof: Substituting one-point sets for A in Lemma 4. 1., and if one-point sets are weakly-α-I-closed sets, we see that every weakly-α-I-paracompact space is weakly-α-I- 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  D. Jankovic and T. R. Hamlet, “New Topological from Old via Ideals,” The American Mathematical Monthly, Vol. 97, No. 4, 1990, pp. 295-310. doi:10.2307/2324512  S. G. Grossley and S. K. Hidebrand, “Semi-Closure,” Texas Journal of Science, Vol. 22, No. 2-3, 1971, pp. 99-112.  E. Hatir, A. Keskin and T. Noiri, “A Note on β-I-Sets and Strongly β-I-Continuous Functions,” Acta Mathematica Hungarica, Vol. 108, No. 1-2, 2005, pp. 87-94. doi:10.1007/s10474-005-0210-2  Q. L. Shi, “On the Weakly-α-I Open Sets,” Acta Mathe-matica Hungarica.