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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 270704, 9 pages
http://dx.doi.org/10.1155/2011/270704
Research Article

On Quasi--Confluent Mappings

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM, Selangor Darul Ehsan, Malaysia

Received 4 March 2011; Accepted 28 April 2011

Academic Editor: Christian Corda

Copyright © 2011 Abdo Qahis and Mohd. Salmi Md. Noorani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new class of mappings called quasi--confluent maps, and we study the relation between these mappings, and some other forms of confluent maps. Moreover, we prove several results about some operations on quasi--confluent mappings such as: composition, factorization, pullbacks, and products.

1. Introduction

A generalization of the notion of the classical open sets which has received much attention lately is the so-called -open sets. These sets are characterized as follows [1]: a subset of a topological space is an -open set if and only if for each , there exists such that and is countable. One can then show that the family of all -open subsets of a space , denoted by , forms a topology on finer than . Using this notion of -open sets, one can then define notions such as -compact and -connected sets whose definitions follow closely the definitions of the related classical notions. For example, a space is called -connected provided that is not the union of two disjoint nonempty -open sets. And is said to be -compact if every -open cover of has a finite subcover. For more information regarding these notions and some recent related results, see [24].

Recall that a subset of a space is said to be a continuum if is connected and compact. Using this idea of a continuum, Charatonik introduced and studied the idea of a confluent mapping between topological spaces [5] as follow: A mapping is said to be confluent provided that for each continuum of and for each component of , we have .

In [6], motivated by Charatonik's work, we have introduced the notion of -confluent mappings and studied its basic properties. In particular, we say a space is an -continuum if it is -connected and -compact at the same time, and a subset of a space is said to be -continuum if is -connected and -compact as a subspace of . Moreover, a mapping is said to be -confluent provided that for each -continuum of and for each component of , we have .

In this paper, we are interested in the further generalizations of the work of Charatonik in the context of -open sets and the idea of quasicomponents. Recall that a quasicomponent of space containing a point is the intersection of all nonempty clopen sets in containing [7]. In particular, we will introduce the notion of quasi--confluent maps and study its relation with the classical mappings such as confluent, -confluent, and quasiconfluent maps. We also study operations on such mappings like compositions, pullback of quasi--confluent, factorizations, and products.

2. Quasi--Confluent Mappings

In this section, we introduce and study a new form of -confluent mapping, which is a quasi--confluent mapping. Throughout this paper, all mappings are assumed to be continuous.

Now, we introduce the following notion.

Definition 2.1. A mapping is said to be quasi--confluent (resp., quasiconfluent) if for each -continuum (resp., continuum) in and for each quasicomponent of , we have .

First, we need the following theorem.

Theorem 2.2 (see [6]). Let be a topological space. Then, (1)every -connected subset of is connected, (2)every -compact subset of is compact,(3)every -continuum subset of is a continuum.

Proposition 2.3. (1) Every -confluent mapping is quasi- -confluent.
(2) Every quasiconfluent mapping is quasi--confluent.

Proof. (1) Suppose that be an -confluent mapping, let be any -continuum in , and let be any point in and be the quasicomponent of in . Then, any component of in contained in the quasicomponents , or . Thus, . Since is an -confluent, then . This implies, . But we have . So, . Thus, . Therefore, is quasi--confluent mapping.
(2) Let be any -continuum in and be any quasicomponent of . Then, is a continuum in by the Theorem 2.2(3). Since, is quasiconfluent. So that, . Thus, is quasi--confluent mapping.

Remark 2.4. It is clear that every -confluent (confluent or quasiconfluent) mapping is quasi--confluent, but the converses are not necessarily true, as shown by the following examples.

Example 2.5. Let , .
(a) Let subspaces of under the usual topology , and , with the topology . Let be the mapping defined by Then, is quasi--confluent but not quasiconfluent. Since, if we take the continuum in , then the quasicomponents of are and . So, , and .
(b) Let subspaces of under the usual topology , and , with the topology . Let be the mapping defined by Then, is quasi--confluent, but not confluent. Since if we take the continuum in , then the components of are and . So, , and .

Example 2.6. Let and with topologies and defined on and , respectively. Let be a mapping defined by , , . Then, is quasi-- confluent, but it is not confluent.

Remark 2.7. Quasi--confluent does not imply -confluent in general, since the quasicomponent containing , may be different from the component containing , , as the following example shows.

Example 2.8. Let be a subspaces of under the usual topology , where be as in Example 2.5, and let with the topology . Let be the mapping defined by Then, is quasi--confluent, but is not -confluent. Note that if we take the -continuum , then the components of are and . Thus, and .

The following diagram summarizes the relations between confluent mapping, quasiconfluent mapping, and -confluent mapping with quasi--confluent mapping.

270704.fig.001

The following theorem shows that under certain conditions, quasi--confluent mapping will be -confluent.

Theorem 2.9. Every quasi--confluent mapping of a compact Hausdorff space into a Hausdorff space is -confluent.

Proof. Let be any -continuum in and any component of . Then, by the Theorem 2.2, is continuum subset of . Since is Hausdorff, then is closed in and since is continuous, then is closed in , since is compact Hausdorff space, so that is compact Hausdorff space. Thus, the quasicomponents are connected and coincide with components of . Thus, . Therefore, is -confluent.

Proposition 2.10. If is hereditarily locally connected, then any quasi--confluent mapping is -confluent.

Proof. It follows that from the fact that in locally connected space, the components and quasicomponents are the same.

Definition 2.11 (see [2]). A space is said to be -space if every -open set is open in .
It is easy to see that in an -space that the continuum and -continuum sets coincide.

Proposition 2.12. If is an -space and if is a mapping of a compact Hausdorff space into a Hausdorff space , then the following are equivalent: (1) is confluent,(2) is -confluent, (3) is quasiconfluent, (4) is quasi--confluent.

Proof. (1) (2). Obvious.
(2) (3). Let be an -confluent mapping, any continuum in , and any quasicomponent of . Since is an -space, then is an -continuum, since is Hausdorff and is compact Hausdorff, so that the components and quasicomponents of are the same. Hence, by assumption. Thus, is quasiconfluent mapping.
(3) (4). It follows from Proposition 2.3(2).
(4) (1). Let is quasi--confluent mapping, any continuum, and be an arbitrary component of , since is an -space, then is an -continuum in , since is a compact Hausdorff and is a Hausdorff. Then, is a quasicomponent of . Thus, . Therefore, is confluent mapping.

Theorem 2.13. Let be a mapping of -dimensional space into space . Then, the following are equivalent: (1) is an -confluent, (2) is quasi--confluent.

Proof. (1) (2). Obvious.
(2) (1). Let be quasi--confluent mapping, any -continuum, and any component of . Since is a zero-dimensional space, then it is totally disconnected. Then the components of are coincide with quasicomponents. Thus, is a quasicomponent of . Then, , by the assumption. Therefore, is an -confluent.

Proposition 2.14. Let be a mapping of space into -dimensional space . Then, the following are equivalent: (1) is quasiconfluent,(2) is quasi--confluent.

Proof. (1) (2). It follows immediately from the Proposition 2.3(2).
(2) (1). Let be any -continuum, and let be any quasicomponent of . Since is zero-dimensional space. Then, the connected subsets of are precisely the singleton sets. Thus, the -continuum are coincide with continuum sets in , therefore, is a continuum in , so that . Hence, is quasiconfluent mapping.

Proposition 2.15. Let be any mapping. If is a hereditarily locally connected space, then the following conditions (1) and (2) are equivalent, and the conditions (3) and (4) are equivalent: (1) is -confluent mapping,(2) quasi--confluent mapping, (3) is confluent mapping,(4) quasiconfluent mapping.

Proof. Similar to the proof of Proposition 2.10.

3. Composition and Factorization of Quasi--Confluent Mappings

In this section, we study the composition and factorization of quasi--confluent mapping. So, we need to recall the following theorem.

Theorem 3.1 (see [6]). Let and be two -confluent mappings, where is a surjective. Then, is an -confluent mapping.

Theorem 3.2. Let be a surjective quasi--confluent of compact Hausdorff space into space and a quasi--confluent of space into Hausdorff space . Then, is quasi--confluent mapping.

Proof. Since and are two compact Hausdorff spaces and since and are two quasi--confluent mappings, then and are -confluent mappings by Theorem 2.9. Therefore, is an -confluent mapping by Theorem 3.1. Then, from Proposition 2.3, is quasi--confluent mapping.

Proposition 3.3. If is hereditarily locally connected space and if and are two quasi--confluent mapping such that is onto closed or open map, then is quasi--confluent mapping.

Proof. The proof follows immediately from Proposition 2.10 and Theorem 3.1.

Theorem 3.4. Let be a mapping of strongly connected space into Hausdorff space , and let be a canonical decomposition () of the following mappings: where is the quotient surjection map, is the inclusion map, and is the bijection mapping, where denote to quotients space over the kernel relation . Then, is a canonical decomposition of -confluent mappings.

Proof. We have to prove that these mappings , , and are -confluent mappings. Let be any arbitrary -continuum in the quotients space and any component of . Since is continuous mapping, then is a Hausdorff, so that is closed in . Then, by the continuity of , we have is closed in . But is strongly connected. Therefore, is connected. This means . So, . Thus, is an -confluent mapping.
It is clearly that and are -confluent mappings, since is a Hausdorff, then the subspace is Hausdorff, and since is strongly connected, then is strongly connected and also Hausdorff. Thus, and the inclusion map are -confluent. Hence, is canonical decomposition of -confluent mappings.

Remark 3.5. In the above theorem, if is strongly connected compact Hausdorff space, then the mapping is the canonical decomposition of quasi--confluent mappings.

Corollary 3.6. If , , and are Hausdorff spaces, is a compact space, and if is a surjective -confluent mapping and is a quasi--confluent mapping, then is -confluent mapping.

Corollary 3.7. If ,, and are Hausdorff spaces, is a compact space, and if is a surjective quasi--confluent mapping and is -confluent mapping, then is a quasi-- confluent mapping.

Now, we study Whyburn's factorization theorem for quasi--confluent mappings. Thus, we recall the definition of a factorable mapping.

Definition 3.8 (see [8]). If be a mapping, any representation of in the form , where and are two mappings and is a certain space, will said to be factorization of , and is said be a factorable mapping and a middle space.

Before we study the factorization property, we state the following theorem.

Theorem 3.9 (see [6]). If is an -confluent of strongly connected compact space into Hausdorff space , then there exists a unique factorization for into two -confluent mappings such that is confluent mapping.

Now, we can get the factorization of a quasi--confluent mapping in the following proposition.

Proposition 3.10. If be a quasi--confluent of strongly connected compact Hausdorff space into Hausdorff space , then there exists a unique factorization for into two quasi--confluent mappings in the form .

Proof. Since and are two quasi--confluent mappings and since is strongly connected compact Hausdorff space and is a Hausdorff space, then from Theorem 2.9 and are -confluent mappings. Thus, has unique factorization in the form by Theorem 3.9.

Next, we study the product property of quasiconfluent mappings.

Let and be any two families of topological spaces. The product space of and is denoted by and , respectively. Let be a mapping for each . Let be the product mappings as follows: for each . The projection of and onto and , respectively, is denoted by and .

Before we get the following result, we need to state the following theorem.

Theorem 3.11 (see [6]). Let be an -confluent mapping, for each of space into Hausdorff space . Then, is an -confluent mapping if the following equality holds:

As immediate consequence of the above theorem, we get the following corollary.

Corollary 3.12. Let be a quasi--confluent mapping of compact Hausdorff space into Hausdorff space for each , then is quasi--confluent mapping if the following equality holds:

Proof. Since, is compact Hausdorff, then the product space is compact Hausdorff, and since is Hausdorff, then the product space is also Hausdorff. From Theorem 2.9, we infer that is an -confluent for each . Then, by Theorem 3.11, is an -confluent mapping. Therefore, is quasi--confluent by Proposition 2.3.

4. Pullback of Quasi--Confluent Mappings

In this section, we study the pullback of quasi--confluent mappings. So, we recall the following definitions.

Definition 4.1 (see [9]). A fiber structure is a triple consisting of two spaces and and a mapping . The space is said to be the fibered (or, total) space, is termed the projection, and is the base space. Next, we recall the definition of the pullback.

Definition 4.2 (see [9]). Let be a fiber structure. Let be any space, and let be any mapping into the base . Let be a subspace of the cartesian product , where , and let be the projection of onto such that ,. The fiber structure is said to be the fiber structure over induced by the mapping , and the projection is said to be the pullback of by .

Now, let be the projection such that , .

We observe that the following diagram is commutative.

270704.fig.002

Before we prove the main results in this section, we state the following lemma.

Lemma 4.3 (see [6]). Let be a mapping, let be any space, and let be any mapping, and if , then , where is the pullback of by .

Theorem 4.4. The pullback of a quasiconfluent mapping is quasi--confluent.

Proof. Let be a quasiconfluent mapping, let be any space, and let be any mapping. Let be any -continuum and any quasicomponent of . Then, is a quasicomponent of by Lemma 4.3. Since every -continuum is continuum, then is a continuum by Theorem 2.2. Thus, is continuum in . Since is quasiconfluent mapping, then for each quasicomponent of . Since , so such that for some quasicomponent of . Thus, . Therefore, is quasi--confluent.

The pullback of quasi--confluent mapping is not necessarily quasi--confluent as shown by the following example.

Example 4.5. Let be the real number with upper limit topology, with the topology , and with topology .
Let be a mapping defined by and let be a mapping defined by Let be a subspace of the cartesian product , where Then, the pullback of by is the projection which is defined by

We note that is quasi--confluent mapping, but is not quasi--confluent mapping.

Since if we take the -continuum , then by Lemma 4.3, we get . But is not -continuum in .

Under certain condition, the pullback of quasi--confluent mapping will be quasi--confluent as shown by the following theorem.

Theorem 4.6. If is a zero -dimensional space and if is a quasi--confluent mapping, then the pullback of is quasi--confluent.

Proof. Let be a quasi--confluent mapping, let be any space, and let be an mapping. Let be any -continuum in , and let be any quasicomponent of , where is the pullback of by . Then is the quasicomponent of by Lemma 4.3. By Theorem 2.2, is continuum. Thus, is continuum in by the continuity of . Since is -dimensional space, then the quasiconfluent mapping equivalent to the quasi--confluent by Proposition 2.14. This implies the continuum and -continuum sets coincide in . Thus, is an -continuum in . Since is a quasi--confluent, then for each quasicomponents of , and since . such that for some quasicomponent of . Thus, . Therefore, is a quasi--confluent.

Corollary 4.7. If is a quasi- -confluent mapping of space into -space , then the pullback of is quasi--confluent mapping.

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