A Note on Weakly-α-I-Functions and Weakly-α-I-Paracompact Spaces *

This paper introduces the new notion of weakly-α-I-functions and weakly-α-I-paracompact spaces in the ideal topological space. And it obtains that some properties of them.


Introduction
Throughout this paper,

 
Cl A and

 
Int A 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 .Let be a subset of a topological space S   , X  .The complement of a semi-open set is said to be semiclosed [2].The intersection of all semi-closed sets containing , denoted by S   sCl S is called the semi-closure [3]
(2) For any x X  and each (3) The inverse image of each closed set in is weaklyα-I-closed.
, where x U is weakly- an open cover of X .Then is weakly-α-I-continuous if and only if the restriction On the other hand,

Sufficiency. Let be any open set of
for each x X  , is weakly-α-I-continuous.
Proof: Necessity.Suppose that f is weakly-α-I-continuous.Let . This show that g is weakly- and by the weaklyα-I-continuous of g ,there exists a weakly-α-I- Therefore we obtain   Similar argument holds for a weakly-α-I-open mapping.

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 y X , there exist disjoint weakly-α-I-open sets and in U V X such that x U  and y V  , and U V    .Definition 3.2.A space X is said to be weakly-α-Iregular space, if for every x X  and every weakly-α-Iclosed set F X  such that x F   , there exist weakly- A space X is said to be weakly-α-Inormal space, if for every pair of disjoint weakly-α-Iclosed sets A , , there exist weakly-α-I-open sets , such that Definition 3.4.An ideal topological space   , , X I  is said to be a weakly-α-I-compact space if every weaklyα-I-open cover of X has a finite subcover.U  , 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 intersection 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 B X .If for every x B  there exist weakly-α-I-open sets Then there also exist weakly-α-I-open sets , such that A U  , and is any family of weakly-α-Iclosed sets which has finite intersection property, and Thus X F  is a weakly-α-I-open set.Since X is a weakly-α-I-compact space, hence there exist finite , and , a contradiction. 1 is any weakly-α-I-open cover for  X , then is a weakly-α-I-closed family that satisfies . So dose not  satisfy finite intersection property, which means has a finite subfamily Since is a weakly-α-I-compact space, the weakly- is a weakly-α-I-open cover of the weakly-α-I-paracompact space X , so that it has a locally finite weakly-α , , then for any .Theorem 3.3.Assuming X is a weakly-α-I-paracompact space, if one-point sets of X are weakly-α-Iclosed 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-Iregular.Applying Lemma 3.1.again we have the conclusion.

Summary
Combining the topological structure with other mathematical 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.
local function of A with respect to I and  [1].It is well known that


of .The semi-interior of , denoted by S S   sInt 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 weaklyα-I-open and weakly-α-I-functions, then obtain some properties of them.First we recall some definitions used in the sequel.Definition 1.1.[4] A subset A of an ideal topological space  , ,  X I  is said to be weakly-α-I-open, if 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  , ,  X I  be an ideal topological space.Then all weakly-α-I-open sets constitute a topology 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.Then A is weakly-α-I-open if and only if A is weakly-α-I-open in   x X  and W be any open set of X Y  containing   g x .Then there exists a basic open set such that α-I-open (resp weakly-α-I-closed) mapping.If y Y  and is a weakly-α-I-closed (resp weakly-α-I-open) set of exists a weaklyα-I-closed (resp weakly-α-I-open) subset of containing such that

Definition 3 . 5 .Theorem 3 . 1 .
An ideal topological space   , , X I  is said to be weakly-α-I-paracompact space, if every weakly-α-I-open cover of X has a locally finite weakly-α-Iopen refinement.be weakly-α-I-perfect, if f is weakly-α-I-closed and for any y Y  , is a weakly-α-I-compact subset of X .An ideal topological space     s s S

2 .
Weakly-α-I-compactness is an inverse invariant of weakly-α-I-perfect mapping.Proof: Let : f X Y  be a weakly-α-I-perfect mapping onto a weakly-α-I-compact space Y .Given a weakly-α-I-open cover   s s S U  of the space X.Since f is a weakly-α-I-perfect mapping and for every y U is a weakly-α-I-open from Theorem 1.1.And from Theorem 2.4 there exists a weaklyα-I-open set y V containing such that y is open.Therefore U is a weakly-α-Ialso a weakly-α-I-open set from theorem1.1.and   