A closure operator is an arbitrary set-valued, set-function where is the power set of a non-void set, [3]. Let.

1) Grounded:

2) Expansive:

3) Sub-additive: This axiom implies the Isotony axiom: implies

4) Idempotent:

The structure, where 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.

A closure space satisfying only the grounded and the isotony closure axioms is called an isotonic space [4].

In the dual formulation, a space is isotonic if and only if the interior function satisfies;

1).

2) implies

Let be a closure space. A family of subsets of is called a c-cover of if covers. A closure space is c-compact if every -cover of has a finite subcover, [3].

Various characterizations of connectedness exist in the realm of topological spaces. For instance, [5] described a space to be connected if and only if for every decomposition of into two non-void closed sets and, the condition is satisfied. In other words, is connected if and only if it cannot be represented as a union of two non-empty, disjoint closed sets.

According to [4] an isotonic space is connected if and only if for all -isotonic doubleton spaces, any continuous function is a constant. This characterization of connectedness is an extension of an equivalent definition of connectedness in topological spaces that was given by [2] where is a discrete space.

In topological 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; This is why the above definition is more suitable for isotonic spaces since it doesn’t involve open or closed sets.

The class of perfect mappings is descended from the broader class of bi-quotient maps. A map is called bi-quotient if whenever and is a covering of by open subsets of, then finitely many with cover some neighbourhood of in [7].

Let and be topological spaces and be a mapping. From [8]

1) is called a compact mapping if is a compact subset of for each

2) is called a perfect mapping if it is a closed and compact mapping.

3) is called an open perfect mapping if is an open and perfect mapping.

4) Every perfect mapping is a quotient map, where is a quotient map if is open in whenever is open in.

The class of perfect mappings may be used in substitution 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.

Let and be isotonic spaces and be a continuous surjective mapping. is called a c-compact mapping if is a c-compact subset of.

A continuous surjective mapping between two isotonic spaces is said to be perfect if it is closed and is a c-compact subset of.

These two definitions closely correlate with their definitions in general topological spaces, except for the fact that a different approach is employed while defining the foregoing topological notions.

There are different forms of connectedness and disconnectedness that have been defined, both in topological spaces and in closure spaces. These definitions can be 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 and strong connectedness are invariants of perfect mappings.

Proof: Let be a perfect mapping of a connected isotonic space onto an isotonic space. Since is a surjection, then. Let, be a perfect mapping, where is a -doubleton isotonic space. Therefore, is perfect. Since is connected, is a constant and hence is also a constant function. This implies that is connected.

Let be a Z-connected isotonic space and be an isotonic space. Let be a perfect mapping (f is a continuous surjection). If is a perfect mapping, where is an isotonic space with more than one element, then by Z-connectedness of, the composition is a constant. It then follows that is also a constant and hence Y is Z-connected.

Let be a perfect mapping between isotonic spaces. Suppose is strongly connected and is not strongly connected. Then, there exists a countable collection of pair-wise semi-separated sets such that. The countable collection, is such that

for all. Moreover,. That is, is not strongly connected which is a contradiction. The contrary, is strongly connected, is true.

Theorem: Total disconnectedness and extremal disconnectedness are not invariants of perfect mapping.

Proof: Let be a perfect mapping from the two-point isotonic space to any isotonic space. Clearly, is totally disconnected while doesn’t have to be. Total disconnectedness is not invariant with respect to perfect mappings.

Since every extremally disconnected space is totally disconnected, then it follows that extremal disconnectedness is not an invariant of perfect mappings.

In this section, we localize the concept of connectedness. We say that a topological property holds locally at a point if there exists a neighborhood which has that property, whether the whole space has that property or not.