On Quasi-𝝎-Confluent Mappings

Abstract

We introduce a new class of mappings called quasi-𝜔-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-𝜔-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 𝜔-open sets. These sets are characterized as follows [1]: a subset 𝑊 of a topological space (𝑋,𝜏) is an 𝜔-open set if and only if for each 𝑥∈𝑊, there exists 𝑈∈𝜏 such that 𝑥∈𝑈 and 𝑈−𝑊 is countable. One can then show that the family of all 𝜔-open subsets of a space (𝑋,𝜏), denoted by 𝜏_𝑤, forms a topology on 𝑋 finer than 𝜏. Using this notion of 𝜔-open sets, one can then define notions such as 𝜔-compact and 𝜔-connected sets whose definitions follow closely the definitions of the related classical notions. For example, a space 𝑋 is called 𝜔-connected provided that 𝑋 is not the union of two disjoint nonempty 𝜔-open sets. And 𝑋 is said to be 𝜔-compact if every 𝜔-open cover of 𝑋 has a finite subcover. For more information regarding these notions and some recent related results, see [2–4].

Recall that a subset 𝐾 of a space 𝑋 is said to be a continuum if 𝐾 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 𝑓∶𝑋→𝑌 is said to be confluent provided that for each continuum 𝐾 of 𝑌 and for each component 𝐶 of 𝑓⁻¹(𝐾), we have 𝑓(𝐶)=𝐾.

In [6], motivated by Charatonik's work, we have introduced the notion of 𝜔-confluent mappings and studied its basic properties. In particular, we say a space 𝑋 is an 𝜔-continuum if it is 𝜔-connected and 𝜔-compact at the same time, and a subset 𝐾 of a space 𝑋 is said to be 𝜔-continuum if 𝐾 is 𝜔-connected and 𝜔-compact as a subspace of 𝑋. Moreover, a mapping 𝑓∶𝑋→𝑌 is said to be 𝜔-confluent provided that for each 𝜔-continuum 𝐾 of 𝑌 and for each component 𝐶 of 𝑓⁻¹(𝐾), we have 𝑓(𝐶)=𝐾.

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

2. Quasi-𝜔-Confluent Mappings

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

Now, we introduce the following notion.

Definition 2.1. A mapping 𝑓∶𝑋→𝑌 is said to be quasi-𝜔-confluent (resp., quasiconfluent) if for each 𝜔-continuum (resp., continuum) 𝐾 in 𝑌 and for each quasicomponent 𝑄𝐶 of 𝑓⁻¹(𝐾), we have 𝑓(𝑄𝐶)=𝐾.

First, we need the following theorem.

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

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

Proof. (1) Suppose that 𝑓∶𝑋→𝑌 be an 𝜔-confluent mapping, let 𝐾 be any 𝜔-continuum in 𝑌, and let 𝑥 be any point in 𝑓⁻¹(𝐾) and 𝑄𝐶_𝑥 be the quasicomponent of 𝑥 in 𝑓⁻¹(𝐾). Then, any component 𝐶_𝑥 of 𝑥 in 𝑓⁻¹(𝐾) contained in the quasicomponents 𝑄𝐶_𝑥, or 𝐶_𝑥⊂𝑄𝐶_𝑥. Thus, 𝑓(𝐶_𝑥)⊂𝑓(𝑄𝐶_𝑥). Since 𝑓 is an 𝜔-confluent, then 𝑓(𝐶_𝑥)=𝐾. This implies, 𝐾⊂𝑓(𝑄𝐶_𝑥). But we have 𝑄𝐶_𝑥⊂𝑓⁻¹(𝐾). So, 𝑓(𝑄𝐶_𝑥)⊂𝐾. Thus, 𝑓(𝑄𝐶_𝑥)=𝐾. Therefore, 𝑓 is quasi-𝜔-confluent mapping.
(2) Let 𝐾 be any 𝜔-continuum in 𝑌 and 𝑄𝐶 be any quasicomponent of 𝑓⁻¹(𝐾). Then, 𝐾 is a continuum in 𝑌 by the Theorem 2.2(3). Since, 𝑓 is quasiconfluent. So that, 𝑓(𝑄𝐶)=𝐾. Thus, 𝑓 is quasi-𝜔-confluent mapping.

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

Example 2.5. Let 𝐾={1/𝑛∶𝑛isapositiveinteger}, 𝐷=𝐾×[0,1].
(a) Let 𝑋=𝐷∪{(0,0),(0,1)} subspaces of ℝ² under the usual topology 𝜏_𝑢, and 𝑌={0,1}, with the topology 𝜏_𝑌={𝜙,𝑌}. Let 𝑓∶𝑋→𝑌 be the mapping defined by 𝑓𝑥,𝑦=0,for𝑥,𝑦∈{(0,0),(0,1)},1,for𝑥,𝑦∈{𝑘}×[0,1],foreach𝑘∈𝐾. (2.1) Then, 𝑓 is quasi-𝜔-confluent but not quasiconfluent. Since, if we take the continuum 𝐾={0,1} in 𝑌, then the quasicomponents of 𝑓⁻¹(𝐾) are {(0,0),(0,1)} and 𝐷. So, 𝑓({(0,0),(0,1)})≠𝐾, and 𝑓(𝐷)≠𝐾.
(b) Let 𝑋=𝐷∪{(0,0),(0,1)}∪([0,1]×{0}) subspaces of ℝ² under the usual topology 𝜏_𝑢, and 𝑌={0,1}, with the topology 𝜏_𝑌={𝜙,𝑌}. Let 𝑓∶𝑋→𝑌 be the mapping defined by 𝑓𝑥,𝑦=0,for𝑥,𝑦=(0,1),1,otherwise. (2.2) Then, 𝑓 is quasi-𝜔-confluent, but not confluent. Since if we take the continuum 𝐾={0,1} in 𝑌, then the components of 𝑓⁻¹(𝐾) are {(0,1)} and 𝑋⧵{(0,1)}. So, 𝑓({(0,1)})≠𝐾, and 𝑓(𝑋⧵{(0,1)}≠𝐾.

Example 2.6. Let 𝑋={𝑝,𝑞,𝑟} and 𝑌={𝑎,𝑏,𝑐} with topologies 𝜏={𝜙,𝑋,{𝑝},{𝑞},{𝑝,𝑞}} and 𝜎={𝜙,𝑌,{𝑎}} defined on 𝑋 and 𝑌, respectively. Let 𝑓∶𝑋→𝑌 be a mapping defined by 𝑓(𝑝)=𝑎, 𝑓(𝑞)=𝑏, 𝑓(𝑟)=𝑐. Then, 𝑓 is quasi-𝜔- confluent, but it is not confluent.

Remark 2.7. Quasi-𝜔-confluent does not imply 𝜔-confluent in general, since the quasicomponent containing 𝑝, 𝑄𝐶(𝑋,𝑝) may be different from the component containing 𝑝, 𝐶(𝑋,𝑝), as the following example shows.

Example 2.8. Let 𝑋=𝐾×[0,1]∪{(0,0),(0,1)}∪([0,1]×{0}) be a subspaces of ℝ² under the usual topology 𝜏_𝑢, where 𝐾 be as in Example 2.5, and let 𝑌=[0,1] with the topology 𝜏_ind={𝜙,𝑌,}. Let 𝑓∶𝑋→𝑌 be the mapping defined by 𝑓𝑥,𝑦=𝑥,∀𝑥,𝑦∈𝑋. (2.3) Then, 𝑓 is quasi-𝜔-confluent, but 𝑓 is not 𝜔-confluent. Note that if we take the 𝜔-continuum 𝐾=[0,1], then the components of 𝑓⁻¹(𝐾) are 𝐶₁={(0,1)} and 𝐶₂=𝑋−{(0,1)}. Thus, 𝑓(𝐶₁)≠𝐾 and 𝑓(𝐶₂)=𝐾.

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

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

Theorem 2.9. Every quasi-𝜔-confluent mapping 𝑓∶𝑋→𝑌 of a compact Hausdorff space 𝑋 into a Hausdorff space 𝑌 is 𝜔-confluent.

Proof. Let 𝐾 be any 𝜔-continuum in 𝑌 and 𝐶 any component of 𝑓⁻¹(𝐾). Then, by the Theorem 2.2, 𝐾 is continuum subset of 𝑌. Since 𝑌 is Hausdorff, then 𝐾 is closed in 𝑌 and since 𝑓 is continuous, then 𝑓⁻¹(𝐾) is closed in 𝑋, since 𝑋 is compact Hausdorff space, so that 𝑓⁻¹(𝐾) is compact Hausdorff space. Thus, the quasicomponents are connected and coincide with components of 𝑓⁻¹(𝐾). Thus, 𝑓(𝐶)=𝐾. Therefore, 𝑓 is 𝜔-confluent.

Proposition 2.10. If 𝑋 is hereditarily locally connected, then any quasi-𝜔-confluent mapping 𝑓∶𝑋→𝑌 is 𝜔-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 (𝑋,𝜏) is said to be 𝜔-space if every 𝜔-open set is open in 𝑋.
It is easy to see that in an 𝜔-space that the continuum and 𝜔-continuum sets coincide.

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

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

Theorem 2.13. Let 𝑓∶𝑋→𝑌 be a mapping of zero-dimensional space 𝑋 into space 𝑌. Then, the following are equivalent: (1)𝑓 is an 𝜔-confluent, (2)𝑓 is quasi-𝜔-confluent.

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

Proposition 2.14. Let 𝑓∶𝑋→𝑌 be a mapping of space 𝑋 into zero-dimensional space 𝑌. Then, the following are equivalent: (1)𝑓 is quasiconfluent,(2)𝑓 is quasi-𝜔-confluent.

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

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

Proof. Similar to the proof of Proposition 2.10.

3. Composition and Factorization of Quasi-𝜔-Confluent Mappings

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

Theorem 3.1 (see [6]). Let 𝑓∶𝑋→𝑌 and 𝑔∶𝑌→𝑍 be two 𝜔-confluent mappings, where 𝑓 is a surjective. Then, ℎ=𝑔∘𝑓 is an 𝜔-confluent mapping.

Theorem 3.2. Let 𝑓∶𝑋→𝑌 be a surjective quasi-𝜔-confluent of compact Hausdorff space 𝑋 into space 𝑌 and 𝑔∶𝑌→𝑍 a quasi-𝜔-confluent of space 𝑌 into Hausdorff space 𝑍. Then, ℎ=𝑔∘𝑓 is quasi-𝜔-confluent mapping.

Proof. Since 𝑋 and 𝑌 are two compact Hausdorff spaces and since 𝑓 and 𝑔 are two quasi-𝜔-confluent mappings, then 𝑓 and 𝑔 are 𝜔-confluent mappings by Theorem 2.9. Therefore, ℎ=𝑔∘𝑓 is an 𝜔-confluent mapping by Theorem 3.1. Then, from Proposition 2.3, ℎ=𝑔∘𝑓 is quasi-𝜔-confluent mapping.

Proposition 3.3. If 𝑋 is hereditarily locally connected space and if 𝑓∶𝑋→𝑌 and 𝑔∶𝑌→𝑍 are two quasi-𝜔-confluent mapping such that 𝑓 is onto closed or open map, then ℎ=𝑔∘𝑓 is quasi-𝜔-confluent mapping.

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

Theorem 3.4. Let 𝑓∶𝑋→𝑌 be a mapping of strongly connected space 𝑋 into Hausdorff space 𝑌, and let 𝑓 be a canonical decomposition (𝑓=inc∘𝑓^′∘𝑝_𝑅𝑓) of the following mappings: 𝑓^′∶𝑋𝑅_𝑓⟶𝑓(𝑋),inc∶𝑓(𝑋)⟶𝑌,and𝑝_𝑅𝑓𝑋∶𝑋⟶𝑅_𝑓, (3.1) where 𝑝_𝑅𝑓 is the quotient surjection map, inc is the inclusion map, and 𝑓^′ is the bijection mapping, where 𝑋/𝑅_𝑓 denote to quotients space over the kernel relation 𝑅_𝑓={(𝑥,𝑥^′)∶𝑓(𝑥)=𝑓(𝑥^′)}. Then, 𝑓 is a canonical decomposition of 𝜔-confluent mappings.

Proof. We have to prove that these mappings 𝑝_𝑅𝑓, 𝑖, and 𝑓^′ are 𝜔-confluent mappings. Let 𝐾 be any arbitrary 𝜔-continuum in the quotients space 𝑋/𝑅_𝑓 and 𝐶 any component of 𝑝_𝑅⁻¹_𝑓(𝐾). Since 𝑝_𝑅𝑓 is continuous mapping, then 𝑋/𝑅_𝑓 is a Hausdorff, so that 𝐾 is closed in 𝑋/𝑅_𝑓. Then, by the continuity of 𝑝_𝑅𝑓, we have 𝑝_𝑅⁻¹_𝑓(𝐾) is closed in 𝑋. But 𝑋 is strongly connected. Therefore, 𝑝_𝑅⁻¹_𝑓(𝐾) is connected. This means 𝑝_𝑅⁻¹_𝑓(𝐾)=𝐶. So, 𝑝_𝑅𝑓(𝐶)=𝐾. Thus, 𝑝_𝑅𝑓 is an 𝜔-confluent mapping.
It is clearly that 𝑓^′ and inc are 𝜔-confluent mappings, since 𝑌 is a Hausdorff, then the subspace 𝑓(𝑋) is Hausdorff, and since 𝑋 is strongly connected, then 𝑋/𝑅_𝑓 is strongly connected and also Hausdorff. Thus, 𝑓^′ and the inclusion map inc are 𝜔-confluent. Hence, 𝑓 is canonical decomposition of 𝜔-confluent mappings.

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

Corollary 3.6. If 𝑋, 𝑌, and 𝑍 are Hausdorff spaces, 𝑋 is a compact space, and if 𝑓∶𝑋→𝑌 is a surjective 𝜔-confluent mapping and 𝑔∶𝑌→𝑍 is a quasi-𝜔-confluent mapping, then ℎ=𝑔∘𝑓 is 𝜔-confluent mapping.

Corollary 3.7. If 𝑋,𝑌, and 𝑍 are Hausdorff spaces, 𝑋 is a compact space, and if 𝑓∶𝑋→𝑌 is a surjective quasi-𝜔-confluent mapping and 𝑔∶𝑌→𝑍 is 𝜔-confluent mapping, then ℎ=𝑔∘𝑓 is a quasi-𝜔- confluent mapping.

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

Definition 3.8 (see [8]). If 𝑓∶𝑋→𝑌 be a mapping, any representation of 𝑓 in the form 𝑓=𝑓₂∘𝑓₁, where 𝑓₁∶𝑋→𝑍 and 𝑓₂∶𝑍→𝑌 are two mappings and 𝑍 is a certain space, will said to be factorization of 𝑓, and 𝑓 is said be a factorable mapping and 𝑍 a middle space.

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

Theorem 3.9 (see [6]). If 𝑓∶𝑋→𝑌 is an 𝜔-confluent of strongly connected compact space 𝑋 into Hausdorff space 𝑌, then there exists a unique factorization for 𝑓 into two 𝜔-confluent mappings 𝑓(𝑥)=𝑓₂∘𝑓₁(𝑥),∀𝑥∈𝑋, (3.2) such that 𝑓₁ is confluent mapping.

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

Proposition 3.10. If 𝑓∶𝑋→𝑌 be a quasi-𝜔-confluent of strongly connected compact Hausdorff space 𝑋 into Hausdorff space 𝑌, then there exists a unique factorization for 𝑓 into two quasi-𝜔-confluent mappings in the form 𝑓=𝑓₂∘𝑓₁.

Proof. Since 𝑓 and 𝑔 are two quasi-𝜔-confluent mappings and since 𝑋 is strongly connected compact Hausdorff space and 𝑌 is a Hausdorff space, then from Theorem 2.9𝑓 and 𝑔 are 𝜔-confluent mappings. Thus, 𝑓 has unique factorization in the form 𝑓=𝑓₂∘𝑓₁ by Theorem 3.9.

Next, we study the product property of quasiconfluent mappings.

Let {𝑋_𝑖}_𝑖∈𝐼 and {𝑌_𝑖}_𝑖∈𝐼 be any two families of topological spaces. The product space of {𝑋_𝑖}_𝑖∈𝐼 and {𝑌_𝑖}_𝑖∈𝐼 is denoted by ∏_𝑖∈𝐼𝑋_𝑖 and ∏_𝑖∈𝐼𝑌_𝑖, respectively. Let 𝑓_𝑖∶𝑋_𝑖→𝑌_𝑖 be a mapping for each 𝑖∈𝐼. Let 𝑓∶∏_𝑖∈𝐼𝑋_𝑖→∏_𝑖∈𝐼𝑌_𝑖 be the product mappings as follows: 𝑓((𝑥_𝑖))=(𝑓_𝑖(𝑥_𝑖)) for each (𝑥_𝑖)∈∏_𝑖∈𝐼𝑋_𝑖. The projection of ∏_𝑖∈𝐼𝑋_𝑖 and ∏_𝑖∈𝐼𝑌_𝑖 onto 𝑋_𝑖 and 𝑌_𝑖, respectively, is denoted by 𝑝_𝑖 and 𝑞_𝑖.

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

Theorem 3.11 (see [6]). Let 𝑓_𝑖∶𝑋_𝑖→𝑌_𝑖 be an 𝜔-confluent mapping, for each 𝑖∈𝐼 of space 𝑋_𝑖 into Hausdorff space 𝑌_𝑖. Then, 𝑓∶_𝑖∈𝐼𝑋_𝑖⟶_𝑖∈𝐼𝑌_𝑖 (3.3) is an 𝜔-confluent mapping if the following equality holds: _𝑖∈𝐼𝜏_𝑖𝜔=_𝑖∈𝐼𝜏_𝑖𝜔,∀𝑖∈𝐼. (3.4)

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

Corollary 3.12. Let 𝑓_𝑖∶𝑋_𝑖→𝑌_𝑖 be a quasi-𝜔-confluent mapping of compact Hausdorff space 𝑋 into Hausdorff space 𝑌 for each 𝑖∈𝐼, then 𝑓∶_𝑖∈𝐼𝑋_𝑖⟶_𝑖∈𝐼𝑌_𝑖 (3.5) is quasi-𝜔-confluent mapping if the following equality holds: _𝑖∈𝐼𝜏_𝑖𝜔=_𝑖∈𝐼𝜏_𝑖𝜔,∀𝑖∈𝐼. (3.6)

Proof. Since, 𝑋_𝑖 is compact Hausdorff, then the product space ∏_𝑖∈𝐼𝑋_𝑖 is compact Hausdorff, and since 𝑌_𝑖 is Hausdorff, then the product space ∏_𝑖∈𝐼𝑌_𝑖 is also Hausdorff. From Theorem 2.9, we infer that 𝑓_𝑖∶𝑋_𝑖→𝑌_𝑖 is an 𝜔-confluent for each 𝑖∈𝐼. Then, by Theorem 3.11, 𝑓∶∏_𝑖∈𝐼𝑋_𝑖→∏_𝑖∈𝐼𝑌_𝑖 is an 𝜔-confluent mapping. Therefore, 𝑓 is quasi-𝜔-confluent by Proposition 2.3.

4. Pullback of Quasi-𝜔-Confluent Mappings

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

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

Definition 4.2 (see [9]). Let (𝑋,𝑓,𝑌) be a fiber structure. Let 𝑍 be any space, and let 𝑔∶𝑍→𝑌 be any mapping into the base 𝑌. Let 𝐸_𝑓 be a subspace of the cartesian product 𝑋×𝑍, where 𝐸_𝑓={(𝑥,𝑧)∶𝑓(𝑥)=𝑔(𝑧)}, and let 𝑝∶𝐸_𝑓→𝑍 be the projection of 𝐸_𝑓 onto 𝑍 such that 𝑝(𝑥,𝑧)=𝑧,∀(𝑥,𝑧)∈𝐸_𝑓. The fiber structure (𝐸_𝑓,𝑝,𝑍) is said to be the fiber structure over 𝑍 induced by the mapping 𝑔, and the projection 𝑝 is said to be the pullback of 𝑓 by 𝑔.

Now, let 𝛾∶𝐸_𝑓→𝑋 be the projection such that 𝛾(𝑥,𝑧)=𝑥, ∀(𝑥,𝑧)∈𝐸_𝑓.

We observe that the following diagram is commutative.

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

Lemma 4.3 (see [6]). Let 𝑓∶𝑋→𝑌 be a mapping, let 𝑍 be any space, and let 𝑔∶𝑍→𝑌 be any mapping, and if 𝐾⊆𝑍, then 𝑝⁻¹(𝐾)=𝑓⁻¹(𝑔(𝐾))×𝐾, where 𝑝 is the pullback of 𝑓 by 𝑔.

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

Proof. Let 𝑓∶𝑋→𝑌 be a quasiconfluent mapping, let 𝑍 be any space, and let 𝑔∶𝑍→𝑌 be any mapping. Let 𝐾⊆𝑍 be any 𝜔-continuum and 𝑄𝐶 any quasicomponent of 𝑝⁻¹(𝐾). Then, 𝑄𝐶 is a quasicomponent of 𝑓⁻¹(𝑔(𝐾))×𝐾 by Lemma 4.3. Since every 𝜔-continuum is continuum, then 𝐾 is a continuum by Theorem 2.2. Thus, 𝑔(𝐾) is continuum in 𝑌. Since 𝑓 is quasiconfluent mapping, then 𝑓(𝑄𝐶^′)=𝑔(𝐾) for each quasicomponent 𝑄𝐶^′ of 𝑓⁻¹(𝑔(𝐾)). Since 𝑝⁻¹(𝐾)=𝑓⁻¹(𝑔(𝐾))×𝐾, so 𝐾=𝑝(𝑓⁻¹(𝑔(𝐾))×𝐾)=𝑝(𝑄𝐶^′×𝐾) such that 𝑄𝐶=𝑄𝐶^′×𝐾 for some quasicomponent 𝑄𝐶^′ of 𝑓⁻¹(𝑔(𝐾)). Thus, 𝑝(𝑄𝐶)=𝑃(𝑄𝐶^′×𝐾)=𝐾. Therefore, 𝑝 is quasi-𝜔-confluent.

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

Example 4.5. Let 𝑋=ℝ be the real number with upper limit topology, 𝑌={𝑎,𝑏} with the topology 𝜏_𝑌={𝜙,{𝑎},𝑌}, and 𝑍=ℝ with topology 𝜏_𝑍={𝜙,ℝ,ℝ−{1},ℝ−{2},ℝ−{1,2}}.
Let 𝑓∶𝑋→𝑌 be a mapping defined by 𝑓(𝑥)=𝑎,if𝑥>0,𝑏,if𝑥≤0, (4.1) and let 𝑔∶𝑍→𝑌 be a mapping defined by 𝑔(𝑧)=𝑎,if𝑧∈ℝ−{1,2},𝑏,if𝑧∈{1,2}. (4.2) Let 𝐸_𝑓 be a subspace of the cartesian product 𝑋×𝑍, where 𝐸_𝑓=(𝑥,𝑧)∶𝑓(𝑥)=𝑔(𝑧). (4.3) Then, the pullback of 𝑓 by 𝑔 is the projection 𝑝∶𝐸_𝑓→𝑍 which is defined by 𝑝(𝑥,𝑧)=𝑧,∀(𝑥,𝑧)∈𝐸_𝑓. (4.4)

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

Since if we take the 𝜔-continuum 𝐾=[0,∞)⊂𝑍, then by Lemma 4.3, we get 𝑝⁻¹(𝐾)=𝑓⁻¹(𝑔(𝐾))×𝐾. But 𝑔(𝐾)={𝑎,𝑏} is not 𝜔-continuum in 𝑌.

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

Theorem 4.6. If 𝑌 is a zero -dimensional space and if 𝑓∶𝑋→𝑌 is a quasi-𝜔-confluent mapping, then the pullback 𝑝 of 𝑓 is quasi-𝜔-confluent.

Proof. Let 𝑓∶𝑋→𝑌 be a quasi-𝜔-confluent mapping, let 𝑍 be any space, and let 𝑔∶𝑍→𝑌 be an mapping. Let 𝐾 be any 𝜔-continuum in 𝑍, and let 𝑄𝐶 be any quasicomponent of 𝑝⁻¹(𝐾), where 𝑝 is the pullback of 𝑓 by 𝑔. Then 𝑄𝐶 is the quasicomponent of 𝑓⁻¹(𝑔(𝐾))×𝐾 by Lemma 4.3. By Theorem 2.2, 𝐾 is continuum. Thus, 𝑔(𝐾) is continuum in 𝑌 by the continuity of 𝑔. Since 𝑌 is 𝑧𝑒𝑟𝑜-dimensional space, then the quasiconfluent mapping equivalent to the quasi-𝜔-confluent by Proposition 2.14. This implies the continuum and 𝜔-continuum sets coincide in 𝑌. Thus, 𝑔(𝐾) is an 𝜔-continuum in 𝑌. Since 𝑓 is a quasi-𝜔-confluent, then 𝑓(𝑄𝐶^′)=𝑔(𝐾) for each quasicomponents 𝑄𝐶^′ of 𝑓⁻¹(𝑔(𝐾)), and since 𝑝⁻¹(𝐾)=𝑓⁻¹(𝑔(𝐾))×𝐾. 𝐾=𝑝(𝑓⁻¹(𝑔(𝐾))×𝐾)=𝑝(𝑄𝐶^′×𝐾) such that 𝑄𝐶=𝑄𝐶^′×𝐾 for some quasicomponent of 𝑓⁻¹(𝑔(𝐾)). Thus, 𝑝(𝑄𝐶)=𝑃(𝑄𝐶^′×𝐾)=𝐾. Therefore, 𝑝 is a quasi-𝜔-confluent.

Corollary 4.7. If 𝑓∶𝑋→𝑌 is a quasi- 𝜔-confluent mapping of space 𝑋 into 𝜔-space 𝑌, then the pullback of 𝑓 is quasi-𝜔-confluent mapping.