On the Behavior of Connectedness Properties in Isotonic Spaces under Perfect Mappings

The topological study of connectedness is heavily geometric or visual. Connectedness and connectedness-like properties play an important role in most topological characterization theorems, as well as in the study of obstructions to the extension of functions. In this paper, the behaviour of these properties in the realm of closure spaces is investigated using the class of perfect mappings. A perfect mapping is a type of map under which the image generally inherits the properties of the mapped space. It turns out that the general behaviour of connectedness properties in topological spaces extends to the class of isotone space.


Introduction
The concept of a topological space is generally introduced and studied in terms of the axioms of open sets.However, alternate methods of describing a topology are often used: neighborhood systems, family of closed sets, closure operator and interior operator.Of these, the closure operator-records [1], was axiomated by Kuratowski.
The class of continuous functions forms a very broad spectrum of mappings comprising of different subclasses with varying properties.In the spectrum of continuous functions is the class of perfect mappings which, although weaker than homeomorphisms, provides a general yet satisfactory means of investigating topological invariants and hence the equivalence of topological spaces.
In an attempt to extend the boundaries of topology, [2] has shown that topological spaces do not constitute a natural boundary for the validity of theorems and results in topology.Many results therefore, can be extended to closure spaces where some of the basic axioms in this space can be dropped.Many properties which hold in basic topological spaces hold in spaces possessing the isotonic property.

Closure Operator and Generalized Closure Space
A closure operator is an arbitrary set-valued, set-function satisfies the first three axioms is called a closure space.If in addition the idempotent axiom is satisfied, then the structure is a topological space.

Isotonic Space
A closure space   , X cl satisfying only the grounded and the isotony closure axioms is called an isotonic space [4].In

 
, X cl the dual formulation, a space is isotonic if and only if the interior function X has a finite subcover, [3].

Connectedness
ns of connectedness exist in the Various characterizatio realm of topological spaces.For instance, [5]  T -isotonic d n spaces is a This characterization of conn is an extension of an equivalent definition of connectedness in topological spaces that was given by [2] where Y is a discrete space.
In topological constant.ectedness spaces, connectedness is defined by means of open and closed sets.That definition can only be extended to closure spaces which have the expanding closure axiom;

A cl A
 This is why the above definition is more su otonic spaces since it doesn't involve open or closed sets.itable for is

Total Disconnectedness
The component  is the union of all connected subsets of X co g , ntainin x [5].It is clear, from the fact that the ion of any fa ly of connected subsets having at least one point in common is also con- , [4].Since the c nts of a space ompone X hen every totally disconnected space is 1 .
T are closed, t

Z-Connectedness
edness, according to [6], is ob-The concept of Z-connect tained by replacing the discrete space {0,1} in the characterization of connectedness, by some other space Z .
Let   , Z cl be an isotonic space with more than ne o element.An isotonic space   , X cl is called Z-con- nected if and only if an tinuous function :

Strongly Connected
An isotonic space   The clas ct mappings ay b s of perfe m e used in subs tio lts of this work.titun of homeomorphisms to investigate the invariance of topological properties.By investigating topological properties in isotonic spaces under the class of perfect mappings, which is a more general class of functions than homeomorphisms, the results and theorems obtained become a pointer of how general the concepts of topology can get.

Main Results
This section summarizes the resu

Perfect Mappings on Isotonic Spaces
Let X and Y be isotonic spaces and : two definitions closely co late These rre with t ni

Invariance of Connec edness Conditions
con-heir defitions in general topological spaces, except for the fact that a different approach is employed while defining the foregoing topological notions.

t
There are different forms of connectedness and dis nectedness that have been defined, both in topological spaces and in closure spaces.These definitions can be and strong co found under Section 2.3 of this paper.The next two theorems describe the behavior of the different forms of connectedness with respect to perfect mappings.
Theorem: Connectedness, Z-connectedness nnectedness are invariants of perfect mappings.Proof: be a perfect mapping of a connected isotonic space X onto an isotonic space Y .Since f is a surject , then , be a perfect mapp X  ing, where   0,1 leton isotonic space.There is a - ant and hence g is also a constant func- is a perfect mapping, where   , Z cl is an isot with more than one element, Z-connectedness of onic space then by X , the composition : gof X Z  is a constant.It then lows that fol g is ant and hence Y is Z-connected.
Let : also a const f X Y  paces.Sup be a perfect mapping between isotonic s pose X is strongly connected and  .X is said to be lo-cally connected if it is locally connected at each of its points.This definition follows directly from general topological spaces.
Theorem: Let : f X Y  be a perfect map of a locally connected isotonic space onto an isotonic space .Then is locally connected as well.
and since is locally connected, then for every , there exists a connected neighborhood  .Thus we have for every f N  y  there exists This result shows that local connectedness is an in-variant of perfect mappings, despite the fact that the property is not an invariant of continuous functions.

Conclusion
The results obtained in this paper show that the invariance of connectedness properties in isotonic spaces under continuous functions extends to perfect mappings.