Classification of Weak Continuities and Decomposition of Continuity

We first introduce 20 weak forms of continuity, which are closely related to 5 known weak forms of continuity. Then we classify them into 9 groups and give 12 decompositions of continuity. 1. Introduction. Continuity is one of the most important concepts in mathematics. In order to find deep properties of continuity, many weak forms of continuity were introduced in the literature. For instance, we have Levine's weak continuity [2] and semicontinuity [3], M. K. Singal and A. R. Singal's almost continuity [6], Husain's almost continuity [1], α-continuity of Mashhour et al. [4], and many others. Each of the above forms of continuity is strictly weaker than continuity. Theoretically, for each weak form of continuity, there is another weak form of continuity such that both of them imply continuity. In this connection, there is one result [3] for the general case. A special case is discussed in [7]. In this note, we develop these results. We introduce 20 weak forms of continuity, which are closely related with the above-mentioned weak continuities. Then we classify them into 9 groups and give 12 decompositions of continuity .


Introduction. Continuity is one of the most important concepts in mathematics.
In order to find deep properties of continuity, many weak forms of continuity were introduced in the literature.For instance, we have Levine's weak continuity [2] and semicontinuity [3], M. K. Singal and A. R. Singal's almost continuity [6], Husain's almost continuity [1], α-continuity of Mashhour et al. [4], and many others.Each of the above forms of continuity is strictly weaker than continuity.Theoretically, for each weak form of continuity, there is another weak form of continuity such that both of them imply continuity.In this connection, there is one result [3] for the general case.A special case is discussed in [7].In this note, we develop these results.We introduce 20 weak forms of continuity, which are closely related with the above-mentioned weak continuities.Then we classify them into 9 groups and give 12 decompositions of continuity.

Preliminaries.
We recall some known definitions.Definition 2.1 [3].A subset S in a topological space X is said to be semiopen if Definition 2.4 [5].A subset S in a topological space is said to be an α-set if S ⊂ int cl int S. Definition 2.5 [4].A mapping f : There are two different definitions of almost continuous mappings, one is given by Husain [1]; the other one is given by M. K. Singal and A. R. Singal [6].In this note, following Mashhour et al. [4], we use precontinuity for Husain's almost continuity, and use almost continuity particularly for M. K. Singal and A. R. Singal's.Definition 2.6 [1].A mapping f : X → Y is said to be precontinuous if for each x ∈ X and each open set Definition 2.7 [6].A mapping f : X → Y is said to be almost continuous if for each The relations of the above five weak forms of continuity are as follows [4]: Our classification is based on Lemma 3.1.
In the following group of weak continuities, (i) is trivial, (ii) and (iii) are given in [4].
Definition 3.2.Let f : X → Y be a mapping and let V be an arbitrary open set in Y .
In the above definitions, (ii) and (iii) are new.
It is known [6] that a mapping f : In the above definitions, (ii) and (iii) are new.
It is easily seen that a mapping f : In the above definitions, (ii) and (iii) are new.The following definitions are all new.
Definition 3.6.Let f : X → Y be a mapping and let V be an arbitrary open set in Y .
The following chart gives the relationships of all the weak forms of continuity in this section: It is easily seen that any mapping is a continuous # mapping.We have the following group of definitions corresponding to Definition 3.3.
We have the following group of definitions corresponding to Definition 3.4.
Now we go to the last group of definitions.
The following chart gives the relationships of all the weak continuities in this section:

Decompositions of continuity.
We need an important lemma.
Lemma 5.1.Let α : 2 X → 2 X be a mapping with α(A∩B) ⊂ αA∩αB and let β : 2 X → 2 X be another mapping with We have proved that f −1 (V ) is an open set, hence f is continuous.(xiii) f is pre-continuous and α # -continuous.

Now we turn to the decomposition of continuity. Because int(A
In the above twelve nontrivial decompositions, if we choose a proper operator β other than identity mapping, cl or int cl, we can have infinitely many decompositions.For instance, we may let βA = A ∪ E, where E is a subset of X such that A ∩ E ≠ φ.

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3 .Lemma 3 . 1 .
Classification of some weak continuities.The following results are known.Let S be a subset in a topological space X.Then (i) int int S = int S; (ii) cl cl S = cl S; (iii) int cl int cl S = int cl S; (iv) cl int cl int S = cl int S.

Definition 3 . 3 .
From this we have the following group of definitions.Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then (i) f is weakly continuous if and only

Definition 3 . 4 .
From this we have the following group of definitions.Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then (i) f is almost continuous if and only

Definition 3 . 5 .
From this we have the following group of definitions.Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then

4 . 2 . 4 . 1 .
Classification of relative continuities.Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then f is continuous if and only if f −1 (V ) is an open set in X.If we relax the requirement on f −1 (V ) from being open in X to being open in a subspace, then we can obtain many new weak forms of continuity.For instance, we have the following group of weak continuities corresponding to Definition 3.Definition Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then

Definition 4 . 2 .
Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then

Definition 4 . 3 .
Let f : X → Y be a mapping and let V be an arbitrary open set in Y .Then

First
Round of ReviewsMarch 1, 2009 Therefore we have the following theorem.Let f : X → Y be a mapping.Then f is continuous if and only if (i) f is continuous and continuous # ; (ii) f is precontinuous and pre (vii) f is almost continuous and almost # continuous; (viii) f is pre-almost continuous and pre-almost # continuous; (ix) f is α-almost continuous and α-almost # continuous.In the above decompositions, (i) is trivial and the other eight are all new.Since int cl int cl f −1 (βV ) = int cl f −1 (βV ) and cl int cl(A∩B) = cl int cl A∩cl int cl B, we have the following decompositions.Let f : X → Y be a mapping.Then f is continuous if and only if (x) f is pre-continuous and pre-semi # -continuous; (xi) f is pre-weakly continuous and pre-weak-semi # -continuous; (xii) f is pre-almost continuous and pre-almost-semi # -continuous; # -continuous;(iii) f is α-continuous and α # -continuous; (iv) f is weakly continuous and weak # continuous; (v) f is pre-weakly continuous and pre-weak # continuous; (vi) f is α-weakly continuous and α-weak # continuous;