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 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 semi-open set is said to be semi- closed [2]. The intersection of all semi-closed sets con- taining , denoted by S Cl S is called the semi-clo- sure [3] of . The semi-interior of , denoted by S S Int S, is defined by the union of all semi-open sets contained in . S In 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. [4] A subset of an ideal topological space ,, I is said to be weakly-α-I-open, if AsClIntClIntA . Definition 1.2. [4] A subset of an ideal topological space S ,, I is said to be a weakly-α-I-closed set if its complement is a weakly-α-I-open set. Theorem 1.1. [4] Let ,, I be an ideal top ological space. Then all weakly-α-I-open sets constitute a topo- logy of . Then (1) and are weakly-α-I-open sets. (2) The finite intersection of weakly-α-I-open sets are weakly-α-I-open sets. (3) If is weakly-α-I-open for each , then is weakly-α-I-open. Theorem 1.2. [4] Let ,, I be an ideal topological space and AU . Then is weakly-α-I-open if and only if is weakly-α-I-open in ,, UU UI . 2. Weakly-α-I-Functions Definition 2.1. A function :,, ,fXI Y 1 is said to be weakly-α-I-continuous , if V is weakly- α-I-open in ,, I , for any ,VY . Definition 2.2. A mapping ,,:,, XIYJ 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) U is weakly-α-I-open (resp. U is weakly-α-I- closed). Theorem 2.1. For a function :,, ,fXI Y , the followings are equivalent: (1) is weakly-α-I-continuous. (2) For any X and each V containing x, there exists weakly-α-I-open containing U such that UV. (3) The inverse image of each closed set in is weakly- α-I-closed. Proof: 13 it is obviously. 12 For each X and each weakly-α-I- -open ,VY containing x. Since is weakly- α-I-continuous, then 1 V is a weakly-α-I-open 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 weakly-α-I-open U containing such that x U 1 V and . 1V x Uf We have 1 xf V V U , where U is weakly- α-I-open for any 1 fG. Thus is weakly- α-I-open by Theorem 1.1. And conclude that is weakly- α-I-continuous. fV f 1 Theorem 2.2. Let :,, ,fXI Y be a fun- ction and U an open cover of . Then is weakly-α-I-continuous if and only if the restriction f :,, , UUU fU IY is weakly-α-I-continuous for each . Proof: Necessity. Let be any open set of V ,Y . Since is weakly-α-I-continuous, 1 V is a weakly- α-I-open set of ,, I . Since U , from Theorem 1.2 is weakly-α-I-open in 1 UfV ,, UU UI . On the other hand, 11 U VU fV and 1 U V is weakly- α-I-open in ,, UU UI . This shows that U is weakly-α-I-continuous for each . Sufficiency. Let be any open set of V ,Y . Since U is weakly-α-I-continuous for each . 1 U V is weakly-α-I- 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 weakly-α-I-conti- nuous. Theorem 2.3. A function is weakly-α-I-continuous if and only if the graph function : XXY defined by , xxfx for each X, is weakly-α-I-continuous. Proof: Necessity. Suppose that is weakly-α-I- 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 weakly-α-I-continuous, there exists a weakly-α-I-open set 0 of U containing such that 0 U U V. By Theorem 1.1. is weakly-α-I-open and 0 U 0 UU UV W. This show that is weakly- α-I-continuous. Sufficiency. Suppose that is weakly-α-I-continuous. Let X and V be any open set of containing Y x. Then V is open in Y and by the weaklyα-I-continuous of ,there exists a weakly-α-I- open set U containing such that UXV. Therefore we obtain UV. This shows that is weakly-α-I-continuous Theorem 2.4. Let :,, ,, XIY J is weakly- α-I-open (resp weakly-α-I-closed) mapping. If Y and is a weakly-α-I-closed (resp weakly-α-I-open) set of U containing 1 y V , then ther e exists a w ea kl y- α-I-closed (resp weakly-α-I-open) subset of con- taining such that Y y 1 VU . Proof: Suppose that is weakly-α-I- c l osed m a ppin g. Given Y and U is a weakly-α-I-open subset of containing 1 y , then U is a weakly-α-I- closed set. Since is weakly-α-I-closed, XU U is weakly-α-I-closed. Hence VY is weakly- α-I-open. It follows fro m X 1f yU that UyfX . Therefore YfXU V and 11 1 VfYfXU XffXUU . Similar argument holds for a weakly-α-I-open map- ping. 3. Weakly-α-I-Paracompact Spaces Definition 3.1. A space is said to be weakly-α-I- Hausdorff, if for each pair of distinct points and in y , there exist disjoint weakly-α-I-open sets and in U V such that U and V, and UV . Definition 3.2. A space is said to be weakly-α-I- regular space, if for every X and every weakly-α-I- closed set X such that F , there exist weakly- α-I-open sets , such that 1 U2 U1 U, 2 U and 12 UU . Definition 3.3. A space is said to be weakly-α-I- normal space, if for every pair of disjoint weakly-α-I- closed sets , , there exist weakly-α-I-op 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 weakly-α-I-compact space if every weakly- α-I-open cover of has a finite subcover. Definition 3.5. An ideal topological space ,, I is said to b e weakly-α-I-paracompact space, if every w eak - ly-α-I-open cover of has a locally finite weakly-α-I- open refinem ent . Definition 3.6. Mapping :,, ,, XIY J is said to be weakly-α-I-perfect, if is weakly-α-I-closed and for any Y , 1 y is a weakly-α-I-compact 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 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 be a weakly-α-I-paracompact space and , a pair of weakly-α-I-closed sets of B . If for every B there exist weakly-α-I-open sets U, V such that U, V and xx UV U V . Then there also exist weakly-α-I-open sets , such that U, and UV . BV Proof: Necessity. If is any family of weakly-α-I- closed sets which has finite intersection property, and F F , then FF FXF X. Thus F is a weakly-α-I-open set. Since is a weakly-α-I-compact 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 weakly-α-I-open cover for , then is a weakly-α-I-closed 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. Weakly-α-I-compactness is an inverse invariant of weakly-α-I-perfect mapping. Proof: Let : XY be a weakly-α-I-perfect map- ping onto a weakly-α-I-compact space Y. Given a weakly-α-I-open cover s S U of the space X. Since is a weakly-α-I-perfect mapping and for every Y choose a finite set , such that y SS 1 y sS y yU U. U is a weakly-α-I-op en from Theorem 1.1. And from Theorem 2.4 there exists a weakly- α-I-open set V containing such that y 11 y yfV U . Since is a weakly-α-I-com - 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 weakly-α-I-open cover of the weakly-α-I-paracompact space , so that it has a locally finite weakly-α-I-open 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 weakly-α-I- open set. 1 sS VW UV is also a weakly-α-I-open set from theorem1.1. and . Theorem 3.3. Assuming is a weakly-α-I-para- compact space, if one-point sets of are weakly-α-I- closed sets, then 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 [1] 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 [2] S. G. Grossley and S. K. Hidebrand, “Semi-Closure,” Texas Journal of Science, Vol. 22, No. 2-3, 1971, pp. 99-112. [3] 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 [4] Q. L. Shi, “On the Weakly-α-I Open Sets,” Acta Mathe- matica Hungarica.
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