WEAKLY CLOSED MULTIFUNCTIONS AND PARACOMPACTNESS

The purpose of the present paper is to investigate weakly closed multifunctions between topological spaces. We discuss basic properties of weakly closed multifunctions and some results of Rose and Jankovi are extended. Our main theorems are concerning some properties of paracompactness under weakly closed multifunctions. These theorems are

Singal and Singal [1] introduced the almost closedness for arbitrary functions between topological spaces.Almost closed functions were also called regular closed by some authors in literature, for example, Noiri  [2].Another important generalization of closedness is the star closedness by Noiri  [3] which implies almost closedness.Recently, Rose and Jankovi: [4] introduced the weak closedness for functions which is a more general concept.Basic properties of weakly closed functions were also investigated in [4].Rose and Jankovi6 [5] discussed conditions on weakly closed functions which assure a Hausdorff range.These results are improvements of Noiri's results in [2].In this paper, weak closedness for multifunctions is studied.Firstly, we give a characterization of weakly closed multifunctions, then some conditions are given such that a weakly closed mutlifunction has a closed graph or strongly closed graph, and related results of [4, 5] are extended.Finally, we investigate some properties of paracompactness under weakly closed multifunctions.Some results of [6][7][8] are generalized.In particular, it is shown that the image of an a-paracompact (resp.a-almost paracompact, a-lll-paracompact, a-almost 'l- paracompact) subset under a weakly closed multifunction is also a-paracompact (resp.a-almost paracompact, a-lll-paracompact, a-almost lll-paracompact) under certain conditions. 2. DEFINITIONS AND PRELIMINARIES.
Let (X,T) be a topoligical space.A C_ X, intA and ctA denote the interior and closure of A respectively.(A) (r(A)) denotes the system of open (closed) neighborhoods of A. z X is in the O-closure of A, i.e., z cloA if each V X;() The family of all regular open subsets of (X,T) is a base for a topology T* on X which is called the semiregularization of T. A C_ X is said to be star closed if A is closed in (X,T*).A(X) is the family of all nonempty subsets of x.For c_ t(X), # {U:U }, and I1 denotes the cardinality of .
Recall that a space x is said to be paracompct if every open covering of x has a locally finite open covering refinement.X is said to be -paracompact if every open covering of cardinality < has a locally finite open refinement.
DEFINITION 1. ([9]) A subset A of a space X is said to be a-parcompact iff for every X-open covering cv. of A there exists an X-locally finite X open covering *" of A which refines .
DEFINITION 2. ([3]) A space x is nearly paracompact iff every regular open cover of X has a locally finite open covering refinement.

DEFINITION 3. ([8])
A subset A of a space X is said to be -nearly parcompact iff every X- regular open cover of A has an X-open X-locally finite refinement which covers A.
DEFINITION 4. ([6, 8]) A space X is almost paracompact (almost -paracompact)iff every open covering of x (every open covering of X of cardinality _< ) has an open locally finite refinement whose union is dense in X. DEFINITION 5. ([8, 10]) A subset A of a space X is said to be a-almost paracompact (a- almost t-paracompct) iff for every X-open cover (X-open cover o and I1 _< ) of A there xists an X-locally finite family of X-open subsets *" which refines and is that the family of X- closures of members of " forms a cover of A. LEMMA 1. ([10]) A space X is almost paracompact if and only if each regular open cover of x has an open locally finite refinement whose union is dense in X.
COROLLARY.Paracompactness implies near paracompactness; near paracompactness implies almost paracompactness.DEFINITION 6. ([11]) A subset A of a space X is said to be a-nearly compact iff every X- regular open cover of A has a finite subcover.DEFINITION 7. A function ]:X-.Y is (1) almost closed if y(C) is closed for each regular closed set C c_ x see 1]; (2) star closed if y(C) is closed for every star closed C c_ X [see 3]; (3) weakly closed if elf(intC) c_ I(C) for every closed C c_ x [see 4, 5].
Clearly we have he following implications: closed-star closed-,almost closedweaklt closed.
A multifunction F:X--,Y is a mapping from X into t(Yl. If AC_X, BC_Y, let F(A) LI {F(x):x _ A}, F + (A) {l .Y:t 6. F(x) for some x A and F(x) if x A}, F-(B)={xa.X:F(x)taB#b}, and F+(B)={x.X:F(x)_B}.Recall that F:X--,Y is upper semicontinuous at X (u.s.c. at X) if F+n(V)_ (x) for every V (F(x)), and F is lower ) on X if it is u.s.c.(1.s.c.) at each x X; and F is continuous if it is both u.s.c, and DEFINITION 8. ([12]) A multifunction F:X-,Y is almost lower semicontinuous at z X (a.l.s.c. at x X) iff cIF-(V) r(x) for each open set V with F(x) t3 V : .F is a.l.s.c, on X if it is a.l.s.c, at each z X.We can prove the following lemma easily.LEMMA 2. The following statements are equivalent for any multifunction F: X-,Y.
(3) F(clV) C_ cIF(U) for all open U C_ X.We say that a multifunction F: DEFINITION 9. ([13]) A multifunction F: X-.Y is said to have a strongly closed graph iff for each pair (z,y).x xY-G(F) there are U E(z) and v E(y) such that (U xclV)faG(F)= .
Similarly to Definition 7, a multifunction F:XY is said to be almost closed (star closed Other basic concepts and terminilogy about topological spaces are referred to Engelking [14].
Let X and Y be topological spaces.Now we shall give a useful characterization of weakly closed multifunctions which is a generalization of a result in [4].
THEOREM 1.The following statements are equivalent for any multifunction F: XY.
(2) For each open U c_ x and any B c_ Y with F-t{B)c_ U, there is an open V c_ Y such that B, c_ V and f-(V) c_ clU.
(3) For each y e Y and each open set U c_ x with F-(y)c_ U, there is V E/(y) such that F-(y) _ ctu.
(3) implies (1).Let C be any closed subset of X, B Y-F(C), then F-()C_ X-C for each yB.Hence there is a Vu(y) such that F-(V)C_cI(X-C).
THEOREM 2. If a weakly closed multifunction F: X--,Y is a.l.s.c., then F is almost closed.
PROOF.Let C be a regular closed subset of x.By the' weak closedness of F, cIF(intC COROLLARY ( [5]).If y:x-Y is a weakly closed and almost continuous function, then/" is almost closed.Now we shall consider conditions under which a weakly closed multifunction has a closed or strongly closed graph.THEOREM 3. Let F:XY be a weakly closed multifunction of a space x into a space Y such that F(y) is O-closed for each y E Y. Then F has a closed graph.
PROOF.Similar to the proof of Theorem 4 of [5].COROLLARY 3.1.([5]) If y:x-Y is a weakly closed function that )'-(y) is O-closed for each y Y, then f has a closed graph.COROLLARY 3.2.Let F:X-,Y be a weakly closed multifunction of a regular space x into a compact space Y such that F-(y) is closed for each y Y. Then F is u.s.c, on X.
PROOF.Since x is regular, then F-l(y) is O-closed for each y 6 Y. Hence, by Theorem 3, F has a closed graph.Now suppose F is not u.s.c, at some point z0 X, there must exist V E(F(0) such that F(U) V for each U E(z0)-Therefore we can choose nets {x,,:a D} c_ X and {v,,:a D} C_ Y-V such that x,--,z 0 and v,, F(z,,) for each D. By the compactness of Y, {v,:a D} must have a cluster point v0 Y-V.Since G(F) is closed, then Vo F(zo).This is a contradiction.THEOREM 4. Let F:X-.Y be a "-open multifunction of a space X into a space Y. Then F has a strongly closed graph if and only if it has a closed graph.PROOF.Necessity is obvious.Sufficiency is similar to the proof of Theorem 3 of [5].COROLLARY 4.1.([5]) Let f:x-Y be a "-open function.Then /" has a strongly closed graph if and only if it has a closed graph.COROLLARY 4.2.Let F:X--*Y be a *-open and weakly closed multifunction such that F-l(V) is O-closed for each v Y. Then F has a strongly closed graph.

MAIN THEOREMS
In this section, we will investigate some properties of paracompactness under weakly closed multifunctions.
THEOREM 5. Let F:X-,Y be a weakly closed, u.s.c., and open multifunction of a space X into a space Y such that F-l(V) is a-nearly compact for each v 6 Y and F(z) is a-paracompact for each 6 X.Then F(K) is a-paracompact if K is an a-paracompact set of X.
PROOF.Let {Uo:a 6 zX be any Y-open cover of F(K).Since F(z) is a-paracompact for each 6 K, then there is a Y-locally finite family of Y-open sets *'x which covers F(z) and refines al. and since F is u.s.c., {F + l(x# ): ( K} is an open cover of K.By the a-paracompactness of K, there is an X-locally finite and x-open cover W'={wa: v} of K which refines {F+I(*'#):z6 K}.Consequently, for each V, there is za6 K such that F(Wa) c_*Ca #.We set (;a {F(Wa)r V:V C,a for each t }, and ( o; a.It is not difficult to see that ( is a Y- open cover of F(K) which refines t.Now we shall show that is Y-locally finite.For each v Y, since *, is X-locally finite, then there exists an x-open set H, containing z such that clHx intersects at most finite members of W" for each z F-a(V).Thus {intclHx:z F-I(v)} is an X- regular open cover of F-I(V) for each v Y. Since F-(V) is a-nearly compact, there exists {,z2, ,zn} C_ F-(y) such that F-(y) c_ o (int clHxi C_ int cl( O Hx,) H u. Consequently, Hu is X-open and intersects at most finite members of W. By the weak closedness of F and Theorem there exists Qu E(y) such that F-(y) c_ F-a(Qu) c_ cIHu c_ el( Hi ).It t'ollows that Qu must meet only with finite members of (.Hence F(K) is a-paracompact.COROLLARY 5.1.If F:X-,Y is a weakly closed, u.s.c., and open multifunction of a paracompact space X onto a space Y such that F-a(y) is a-nearly compact for each y e Y, and F(z) is a-paracompact for each z X, then Y is also paracompact.COROLLARY 5.2.([7]) If F:X-Y is an almost closed, open and u.s.c, multifunction of a paracompact space x onto a space Y such that F-(y) is a-nearly compact for each y Y and F(z) is a-paracompact for each X, then Y is paracompact.paracompact space X onto a space Y such that f-(y) is a-nearly compact for each y Y, then Y is also paracompact.Similarly we can obtain the following series of results.THEOREM 6.Let F:X-,Y be a weakly closed, u.s.c., and open multifunction of a space x into a space Y such that F-(V) is a-nearly compact for each v E Y and F(z) is a-lll-paracompact for each z (; x.Then F(K) is a--paxacompact if K is an a-'l-paracompact subset of X.
COROLLARY.If F:X-,Y is a weakly closed, u.s.c, and open multifunction of an paracompact space X onto a space Y such that F(x) is a-:lll-paracompact for each X and F-(V) is a-nearly compact for each v Y, then Y is -paracompact.
THEOREM 7. Let F:XY be a weakly closed, u.s.c., a.l.s.c, and open multifunction of a space X into a space Y such that F-i(it) is a-nearly compact for each paracompact (a-lll-paracompact) for each q X.Then F(K) is a-almost paracompact (a-almost tll-paracompact) if K is an a-almost paracompact (a-almost -paracompact) subset of X.
COROLLARY ( [6]).If .f:X--Y is a closed, continuous and open function of an almost -paracompact space X onto a space Y such that j'-(v) is compact for each v E Y, then Y is also almost -paracompact.THEOREM 8. Let F:X-Y be a weakly closed, continuous and irreducible multifunction of a space X onto an almost paracompact space Y such that F-(V) is a-nearly paracompact for each it Y and F(z) is a-nearly compact for each z X.Then x is almost paracompact.
PROOF.Let At {U,,:a A be any regular open cover of X.Since F-I(y) is a-nearly ,paxacompact for each It Y, there is an X-locally finite family of X-open sets *'v which covers F-l(V and refines At.By the weak closedness of F and Theorem 1, there is wv E E(it) such that F (it) c_ F I(Wv) C_ cl(,'v # for each It q Y. Thus we obtain an open cover W" {wv: v q Y} of Y. Since Y is almost paracompact, there is a locally finite faanily of open subsets refining such that {cIDa:B q V covers Y. Hence for each B E , there must be v Y such that D _C Wu and then F-I(D) C_ F-l(Wuo) C_ cl(*'a # ).Since F is 1.s.c., {F (Da): B q V is a family of open subsets of X, and then for each BV, F-(Da)=F-(Da)t3cifua # =ci(F-(D,)taV'a#)t3F-(D/)C_cI(F-(Da)taf#).Consequently, we have clF - (D)   Col(F_ -(D)f'l #) for each / V. Now let (={F-(D)V:V_'f'u} for each / V, and (= (.It is evident that ( is a family of open subsets of X which refines At.Firstly, we shall show that ( is locally finite.To do this, for each X, since F(z) is a-nearly compact and is locally finite, there is a regular open G c_ Y with F(z) C_ G and G meets at most finite members of .Then F + I(G) intersects at most finite members of the family {F-I(D):/ V and since F is u.s.c., F + (G) E(z).This shows that ( is locally finite.Now we shall prove that the closures of members of form a covering of x.Since F is weakly closed, we'have clDC_ciF(F-(D)) C_ cIF(int clF-(D/)) C_ F(cIF-(D)) for each / E V. Therefore Y OclD C_ O F(ciF-I(D)) F(UcIF-(D)).Since F is irreducible and {F-(D):/ 6 V is closure-preserving, being locally finite, then UcIF-(D)= X and since each ( v) is closure-preserving, being locally finite, X UciF-(D) C_ Ucl(F-l(Do)'u #) C_ Ucl((/ # {clS:S (} #.Therefore, by Lemma 1, x is an almost paracompact space. COROLLARY ([8]).Let f:X--.Y be a star closed, continuous and irreducible function of a space X onto an almost paracompact space Y such that y-(y) is a-nearly compact for each It Y.
Then X is almost paracompact.
By the similar proof of Theorem 8, we can get the following results.THEOREM 9. Let F:X--Y be a continuous, weakly closed and irreducible multifunction of a space X onto a space Y such that F-(y) is a-'l-paracompact for each It E Y and F(z) is a-nearly compact for each z X.Then X is almost -paracompact if Y is almost -paracompact.
COROLLARY ([6]).If f:x--r is a closed, continuous and irreducible function of a space x onto an almost -paracompact space Y such that f-'(v) is -compact for each v Y. Then X is also almost -paracompact.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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COROLLARY 5 . 3 .
([8])If .:X-Y is an almost closed, open and continuous function of a