CONTRA-CONTINUOUS FUNCTIONS AND STRONGLY S-CLOSED SPACES

. In 1989 Ganster and Reilly [6] introduced and studied the notion of LC-continuous functions via the concept of locally closed sets In this paper we consider a stronger form of LC-continuity called contra-continuity We call a function f (X, 7.) (Y, r) contra-continuous if the preimage of every open set is closed. A space (X, 7) is called strongly S-closed if it has a finite dense subset or equivalently if every cover of (X, 7) by closed sets has a finite subcover. We prove that contra-continuous images of strongly S-closed spaces are compact as well as that contra-continuous, /5-continuous images of S-closed spaces are also compact. We show that every strongly S-closed space satisfies FCC and hence is nearly compact.


INTRODUCTION.
The field of the mathematical science which goes under the name of topology is concerned with all questions directly or indirectly related to continuity. General Topologists have introduced and investigated many different generalizations of continuous functions. One of the most significant of those notions is LC-continuity. Ganster and Reilly [6] defined a function f: (X, 7-) (Y, or) to be LCcontinuous if the preimage of every open set is locally closed. A set A C (X, 7-) is called locally closed [4] FG [25]) if it can be represented as the intersection of an open and a closed set. The importance ofLC-continuity is that it happens to be the dual of near continuity precontinuity) to continuity, i.e. a function f (X, "r) (Y, r) is continuous if and only if it is LC-continuous and nearly continuous [7].
Due to this theorem we can obtain interesting and useful variations of results in functional analysis, for example theorems concerning open mappings and closed graph theorem 17,18,20,27,28] In this paper we present a new generalization of continuity called contra-continuity. We define this class of functions by the requirement that the inverse image of each open set in the codomain is closed in the domain. This notion is a stronger form of LC-continuity This definition enables us to obtain the following results (1) Contra-contlnuous images of spaces having a dense finite subset are compact and (2) Contra-continuous, /5-continuous images of S-closed spaces are compact.
In 1976 Thompson [26] introduced the notion of S-closed spaces via Levine's semi-open sets 12].
A space (X, 7.) is called S-closed if every semi-open cover has a finite subfamily the closures of whose members cover X or equivalently if every regular closed cover has a finite subcover In what turns out 30, the space property of having a finite dense subset is equivalent to the following property Every closed cover has a finite subcover Hence this is a stronger form a S-closedness and we call spaces having this property strongly S-closed Thus restating our result we have Contra-continuous images of strongly Sclosed spaces are compact Moreover we observe that contra-continuity is properly placed between Levine's strong continuity [11] and Ganster and Reilly's LC-continuity [6] In fact it is even a weaker form of Noiri's perfect continuity 16] A decomposition of perfect continuity is presented by showing that a function f" (X, 7-)  In Section 2 we study contra-continuous functions, while Section 3 is devoted to strong Sclosedness. In Section 4 we show that every strongly S-closed space satisfies FCC semi-irreducible) and hence is nearly compact.

CONTRA-CONTINUOUS FUNCTIONS
The following characterization of contra-continuity can be obtained by using the same technique of the similar result involving continuity. (1) f is contra-continuous.
(2) For each z E X and each closed set V m Y wtth f(:v) V, there exists an open set U in X such that z U and f (U) C V.
(3) The inverse image of each closed set in Y is open in X.
Since closed sets are locally closed, then we have TIEOREM 2.2. Every contra-continuous function is LC-continuous.
3. An LC-continuous function need not be contra-continuous. The identity function on the real line with the usual topology is an example of an LC-continuous function (even a continuous function) which is not contra-continuous REMARK 2.4. In fact contra-continuity and continuity are independent notions. Examples 2.3 above shows that continuity does not imply contra-continuity while the reverse is shown in the following example EXAMPLE 2.5. A contra-continuous function need not be continuous Let X {a,b} be the Sierpinski space by setting 7. {3, {a}, X} and a={3,{b},X} The identity functions .f (X, 7-) (X, r) is contra-continuous but not continuous In 1960 Levine [11] defined a function f" (X, 7.)--(Y,a) to be strongly continuous if f(A) c f(A) for every subset A ofX or equivalently if the inverse image of every set in Y is clopen in X 11, Corollary 2]. Thus we have THEOREM 2.6. Every strongly continuous function is contra-contlnuous, l'q In 1984 Noiri [16] introduced the notion of perfect continuity between topological spaces. By indiscrete if and only if every semi-continuous function f" (X, 7.) (Y, r) is perfectly continuous or equivalently if and only if the identity function f" (X, 7-) (X, 7.) is perfectly continuous. In [14] locally indiscrete spaces are called partition spaces. (1) f is perfectly continuous.
(3) f is a-continuous and contra-continuous.
(4) f is nearly continuous and contra-continuous.
In the case when contra-continuity is reduced to LC-continuity we have the following result proved PROOF. Let f (X, 7-) (Y, a) be contra-continuous and nearly continuous and let Xbe almost compact. Let (V,)zei be an open cover of Y. Then (f-l(V,))ze is a closed, nearly open cover ofX due to our assumptions on f By Theorem 2.8 (f-(V,)),s is a clopen cover of X. Since X is almost compact, then for some finite J C I we have X to ,:f-l(V,) to ,af-(V,  COVERING SPACES WITH CLOSED SETS In this section we will give a characterization of spaces, where covers by closed subsets admit finite subcovers. DEFINITION 2. A space is called strongly S-closed if every closed cover ofX has a subcover. REMARK 3.1. Note that the notion of strong S-closedness is independent from compactness. The real line with the cofinite topology is a compact space, which is not strongly S-closed, while again the real line but this time with a topology in which non-void open sets are the ones containing the origin (in such cases the origin is called a generic point 14]) is an example of a non-compact, strongly S-closed space. Thus a T. strongly S-closed space need not be finite. A space is T. if singletons are open or closed. However a Tl-space is strongly S-closed if and only if it is finite. THEOREM 3.2 Every strongly S-closed space X is S-closed. PROOF. It is shown in [8, Theorem 3.2] that X is S-closed if and only if every regular closed cover ofX has a finite subcover. Since every regular closed set is closed the theorem is clear. I-! Note that the reverse in the theorem above is not always true. The real line (in fact any infinite set) with the cofinite topology is an example of an S-closed space, which is not strongly S-closed, since the trivial subsets are the only regular closed sets. THEOREM 3.3. For a space X thefollowing are equivalent: (1) X is strongly S-closed (2) Xhas a finite dense subset. PROOF.
(1) = (2) Since { {x} a: E X} is a closed cover ofX, then for some finite set S C X we have X LI xs {z} S. Thus S is finite and dense in X.
(2) = (1) By (2) for some finite set S {xl, ...,xT} c X we have S X If (A,), is a closed cover of X, then for each index k n there exists an index z(k) E I such that xk E A,k Thus {Xk} C Az(k) and hence X (_J .=l(Xk C U __IA,,.! Thus (A,(1),...,A,(,r)) is a finite subcover of (A,), and so X is strongly S-closed COROLLARY 3.4. If X is strongly S-closed, tken tke set i(X) of all isolated points of X Is fimte The following result can be easily verified Its proof is straightfoward THEOREM 3.5 Strong S-closedness ts open keredttary. [2] Recall that a space (X, 7-) is called d-compact [10] if every cover of Xby dense subsets has a finite subcover THEOREM 3.6. Every d-compact, strongly S-closed space X ts fimte PROOF. By Theorem 3 3 X has a finite dense subset S Since {S U {x) x E X \ S} is a dense cover ofX, then by assumption X \ S is finite Thus X is finite being the union of two finite sets Next we give a condition under which S-closedness implies strong S-closedness. Recall that a set A c X is called a semi-generahzed closed set sg-closed set) [2]   PROOF. Let f (X, 7-) (Y, or) be strongly continuous and onto. Assume that X is strongly S-closed. Let (V,)ei be an open cover of Y Then (f-(V)),e is a closed cover ofX, sincefis contracontinuous. Thus for some finite J C I we have X LI,jf-I(v). Sincefis onto, then Y or equivalently Y is compact. !-i REMARK 3.9 In the theorem above, contra-continuity cannot be reduced to LC-continuity, since there are strongly S-closed spaces which are not compact and since the identity function is always LCcontinuous. In the notion of the above given proof and since regular closed sets are semi-open, then the following result is obvious. THEOREM 3.11. Every contra-contmuous, /-contmuousfunctton ts semt-continuous. In 1968 Singal and Singal [23] introduced the notion of almost-continuity By definition a function f (X, 7-) (Y, or) is called almost-continuous [23] if for each :r X and for each neighborhood V of f(z), there is a neighborhood Uofx such that f(U) c Int V In 1974 Long and Herrington [13] proved Finally we note that in 1980 Hong [9] introduced another stronger form of S-closedness called RScompactness. He defined a space (X, r) to be RS-compact [9] if every cover ofX by regular semi-open sets has a finite subfamily such that the interiors of its members cover X. A set A is called regular semiopen [5]  A space (X, 7-) is nearly compact [22] if every open cover ofXhas a finite subfamily such that the interiors of its closures cover X or equivalently if every cover of X by regular open sets X has a finite subeover.
In this section we will show that the following diagram holds and none of its implications is reversible: (1) SisFCC.   (2)  TItEOREM 4.5. If a space (X, 7-) is FCC, then X is both S-closed and nearly compact Finally we point out that an example of an FCC-space which is not strongly S-closed is an infinite cofinite space. Thus even an irreducible space need not be strongly S-closed. On the other hand a strongly S-closed space need not be irreducible: a finite discrete space with at least two points is an easy example

Call for Papers
Intermodal transport refers to the movement of goods in a single loading unit which uses successive various modes of transport (road, rail, water) without handling the goods during mode transfers. Intermodal transport has become an important policy issue, mainly because it is considered to be one of the means to lower the congestion caused by single-mode road transport and to be more environmentally friendly than the single-mode road transport. Both considerations have been followed by an increase in attention toward intermodal freight transportation research. Various intermodal freight transport decision problems are in demand of mathematical models of supporting them. As the intermodal transport system is more complex than a single-mode system, this fact offers interesting and challenging opportunities to modelers in applied mathematics. This special issue aims to fill in some gaps in the research agenda of decision-making in intermodal transport.
The mathematical models may be of the optimization type or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support decisions on the strategic, tactical, and operational levels. The decision-makers belong to the various players in the intermodal transport world, namely, drayage operators, terminal operators, network operators, or intermodal operators.
Topics of relevance to this type of decision-making both in time horizon as in terms of operators are: • Intermodal terminal design • Infrastructure network configuration • Location of terminals • Cooperation between drayage companies • Allocation of shippers/receivers to a terminal • Pricing strategies • Capacity levels of equipment and labour • Operational routines and lay-out structure • Redistribution of load units, railcars, barges, and so forth • Scheduling of trips or jobs • Allocation of capacity to jobs • Loading orders • Selection of routing and service Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/jamds/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/, according to the following timetable:

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