Homotopy Extension Property in Homotopy Theory for Topological Semigroups

Abstract

The purpose of this paper is to extend the concept of homotopy extension property in homotopy theory for topological spaces to its analogical structure in homotopy theory for topological semigroups. In this extension, we also give some results concerning on absolutely retract and its properties.

1. Introduction

The definition of homotopy theory for topological semigroups and most of the backgrounds for this paper have been worked out previously by Čerin in [1]. He introduced the concepts of 𝑆-homotopy relation, pathwise 𝑆_ℵ-connectedness, 𝑆-homotopy domination, 𝑆-contractibility, and 𝑆_ℵ-fibration.

A topological semigroup (𝐴,∘) is called an 𝑆-subspace of a topological semigroup (𝑆,∗) if 𝐴 is a subspace of 𝑆, the map ∗ takes the product 𝐴×𝐴 into 𝐴 and ∗(𝑥,𝑦)=∘(𝑥,𝑦) for all 𝑥,𝑦∈𝐴. The pair (𝑆^𝐼,𝑝(∗)) is a topological semigroup with the compact-open topology on a path space 𝑆^𝐼, where the multiplication 𝑝(∗)∶𝑆^𝐼×𝑆^𝐼→𝑆^𝐼 on 𝑃𝑎(𝑆) defined by[𝑝(∗)(𝛼,𝛽)](𝑡)=∗(𝛼(𝑡),𝛽(𝑡)),for𝛼,𝛽∈𝑆^𝐼,𝑡∈𝐼.(1.1)

𝑃(𝑆,∗) will denote a topological semigroup (𝑆^𝐼,𝑝(∗)). For 𝑠∈𝑆, 𝑠̃ will denote the constant path into 𝑠 in (𝑆,∗). The function 𝑓∶(𝑆₁,∗)→(𝑆₂,∘) is called an 𝑆-map of (𝑆₁,∗) into (𝑆₂,∘) if 𝑓 is a continuous map of space 𝑆₁ into 𝑆₂ such that𝑓(∗(𝑥,𝑦))=∘(𝑓(𝑥),𝑓(𝑦)),for𝑥,𝑦∈𝑆₁.(1.2)

Two 𝑆-maps 𝑓,𝑔∶(𝑆,∗)→(𝑂,∘) are called 𝑆-homotopic (written 𝑓≃𝑠𝑔) if there is an 𝑆-map 𝐻∶(𝑆,∗)→𝑃(𝑂,∘) (called 𝑆-homotopy) such that 𝐻(𝑠)(0)=𝑓(𝑠) and 𝐻(𝑠)(1)=𝑔(𝑠) for all 𝑠∈𝑆, for more details see [1–3].

Recall [1] that a surjective 𝑆-map 𝑓∶(𝑆,∗)→(𝑂,∘) is called 𝑆_ℵ-fibration if it has an 𝑆-homotopy lifting property for any element in ℵ, where ℵ is the collection of all topological semigroup. Now we introduce the concept of an 𝑆-lifting function of an 𝑆_ℵ-fibration as follows: let 𝑓∶(𝑆,∗)→(𝑂,∘) be an 𝑆-map. Then the function 𝐿_𝑓∶(Δ𝑓,∗×𝑝(∘))→𝑃(𝑆,∗) is called an 𝑆-lifting function for 𝑓 if it is satisfying the following:(1)𝐿_𝑓 is an 𝑆-map,(2)𝐿_𝑓(𝑠,𝛼)(0)=𝑠 for (𝑠,𝛼)∈Δ𝑓,(3)𝑓[𝐿_𝑓(𝑠,𝛼)]=𝛼 for (𝑠,𝛼)∈Δ𝑓,

where Δ𝑓={(𝑠,𝛼)∈𝑆×𝑃𝑎(𝑂)∶𝑓(𝑠)=𝛼(0)}. If 𝐿_𝑓(𝑠,𝑓∘𝑠)̃=𝑠̃ for all 𝑠∈𝑆, then the 𝑆-lifting function is called an 𝑆_ℵ-regular lifting function. Also we say that the 𝑆-map 𝑓∶(𝑆,∗)→(𝑂,∘) is an 𝑆_ℵ-regular fibration if it has 𝑆_ℵ-regular lifting function.

Note that we can easily prove that 𝑆-map 𝑓 is an 𝑆_ℵ-regular fibration if and only if it has 𝑆-regular lifting function. Further, if we let 𝑓∶(𝑆,∗)→(𝑂,∘) be an 𝑆_ℵ-fibration and (𝐸,∘) be an 𝑆-subspace of (𝑂,∘). Then 𝑆-map 𝑓|_𝑓⁻¹_(𝐸)∶(𝑓⁻¹(𝐸),∗)→(𝐸,∘) is an 𝑆_ℵ-fibration, where 𝑓|𝐸 will be the 𝑆_ℵ-fibration 𝑓|_𝑓⁻¹_(𝐸). Thus, one by defining two 𝑆_ℵ-fibrations 𝑓₁∶(𝑆₁,∗₁)→(𝑂,∘) and 𝑓₂∶(𝑆₂,∗₂)→(𝑂,∘), an 𝑆-map ℎ∶(𝑆₁,∗₁)→(𝑆₂,∗₂) and setting the relation 𝑓₂∘ℎ=𝑓₁, then we can also study two 𝑆-fiber maps ℎ∶(𝑆₁,∗₁)→(𝑆₂,∗₂) and 𝑔∶(𝑆₁,∗₁)→(𝑆₂,∗₂) whether they are 𝑆-fiber homotopic.

Theorem 1.1 (See [4]). Let 𝑋, 𝑌, and 𝑍 be topological spaces. Then the map 𝐹∶𝑋×𝑌→𝑍 always gives rise to a map 𝐻∶𝑋→𝐶(𝑌,𝑍) by defining 𝐻(𝑥)(𝑦)=𝐹(𝑥,𝑦) for all 𝑥∈𝑋,𝑦∈𝑌. If 𝑌 is locally compact and regular space, then the map 𝐻∶𝑋→𝐶(𝑌,𝑍) always gives rise to map 𝐹∶𝑋×𝑌→𝑍 by defining 𝐹(𝑥,𝑦)=𝐻(𝑥)(𝑦) for all 𝑥∈𝑋,𝑦∈𝑌.

By the natural topological semigroup, we mean a topological semigroup (𝑆,𝜋_𝑖), where 𝜋_𝑖 is continuous associative multiplication on 𝑆 defined by 𝜋₁(𝑥,𝑦)=𝑥 and 𝜋₂(𝑥,𝑦)=𝑦 for all 𝑥,𝑦∈𝑆.

2. 𝑆-Homotopy Extension Property

In this section, we extend the notions of an absolute retract (AR), an absolute neighborhood retract (ANR) and homotopy extension property in homotopy theory for topological spaces into their analogical structure in homotopy theory for topological semigroups. Also we show the relations among them.

Definition 2.1 (See [1]). Let (𝑂,∗) be an 𝑆-subspace of topological semigroup (𝑆,∗). The 𝑆-retraction of (𝑆,∗) onto (𝑂,∗) is an 𝑆-map 𝑟∶(𝑆,∗)→(𝑂,∗) such that 𝑟(𝑠)=𝑠 for all 𝑠∈𝑂. When there is an 𝑆-retraction of (𝑆,∗) onto (𝑂,∗), then we say that (𝑂,∗) is an 𝑆-retract of (𝑆,∗).

Definition 2.2. A topological semigroup (𝑆,∗) is called an 𝑆_ℵ-absolute retract (𝑆_ℵ-AR) for normal topological semigroups (𝑋,•) in the class ℵ if for every closed 𝑆-subspace (𝐴,•) of (𝑋,•), any 𝑆-map 𝑓∶(𝐴,•)→(𝑆,∗) has an extension 𝑆-map 𝐹∶(𝑋,•)→(𝑆,∗).

Definition 2.3. A topological semigroup (𝑆,∗) is called an 𝑆_ℵ-absolute neighborhood retract (𝑆_ℵ-ANR) for normal topological semigroups (𝑋,•) in the class ℵ if for every closed 𝑆-subspace (𝐴,•) of (𝑋,•), any 𝑆-map 𝑓∶(𝐴,•)→(𝑆,∗) can be extended to an open neighborhood 𝑆-subspace of (𝐴,•).

By using the above Definitions 2.2 and 2.3, we can easily state the following theorem.

Theorem 2.4. If (𝑆,∗) is an 𝑆_ℵ-AR (resp. 𝑆_ℵ-ANR), then 𝑆 is an AR (resp. ANR).

The converse of Theorem 2.4 above needs not to be true for example: If ⋅ denotes the usual multiplication on the the unit closed interval 𝐼. 𝐽 denotes the first half [0,1/2] of 𝐼. It is clear that the topological semigroup (𝐽,⋅) is a closed 𝑆-subspace of a normal topological semigroup (𝐼,⋅). Also it is clear that 𝐽 is an ANR. Take 𝐴∶=𝐽 and 𝑋∶=𝐼 in Definition 2.3, let the identity 𝑆-map 𝑖∶(𝐽,⋅)→(𝐽,⋅) has an extension 𝑆-map 𝑅∶(𝑀,⋅)→(𝐽,⋅), where (𝑀,⋅) is an open neighborhood 𝑆-subspace of (𝐽,⋅) in (𝐼,⋅). In the relative usual topology on 𝐼, we can get an element 𝑥∈𝑀 and 𝑥∉𝐽, that is, 1/2<𝑥≤1. Since 𝑥/2∈𝐽 and an 𝑆-map 𝑅 would have to satisfy an impossible condition:𝑥2=𝑟𝑥2=𝑅𝑥⋅12=𝑅(𝑥)⋅𝑅12=12⋅𝑅(𝑥)(2.1)

that implies 𝑅(𝑥)=𝑥∉𝐽. Hence (𝐽,⋅) is not 𝑆_ℵ-ANR. Similarly, 𝐽 is an AR that but it is not 𝑆_ℵ-AR.

The converses of Theorem 2.4 are true for the collection of natural topological semigroups 𝒩_𝑖 as shown in the following theorem.

Theorem 2.5. A topological semigroup (𝑆,𝜋_𝑖) is an 𝑆_𝒩𝑖-AR (resp. an 𝑆_𝒩𝑖-ANR) for normal topological semigroups (𝑋,𝜋_𝑖) if and only if the topological space 𝑆 is AR (resp. ANR).

Proof. It is clear that the function 𝑓∶(𝑋,𝜋_𝑖)→(𝑆,𝜋_𝑖) is an 𝑆-map if and only if the function 𝑓∶𝑋→𝑆 is a continuous. Hence a topological semigroup (𝑆,𝜋_𝑖) is an 𝑆_𝒩𝑖-AR (resp. an 𝑆_𝒩𝑖-ANR) for normal topological semigroups (𝑋,𝜋_𝑖) if and only if the topological space 𝑆 is AR (resp. ANR).

Theorem 2.6. If(𝑆,∗)is an𝑆_ℵ-AR, then𝑃(𝑆,∗)is also an𝑆_ℵ-AR.

Proof. Let (𝐴,•) be a closed 𝑆-subspace of a normal topological semigroup (𝑋,•) and let 𝐻∶(𝐴,•)→𝑃(𝑆,∗) be an 𝑆-map. Since 𝐼 is a locally compact and regular, then by Theorem 1.1, for 𝑡∈𝐼, we can define an 𝑆-map 𝑓_𝑡∶(𝐴,•)→(𝑆,∗) by 𝑓_𝑡(𝑎)=𝐻(𝑎)(𝑡),for𝑎∈𝐴.(2.2) Since (𝑆,∗) is an 𝑆_ℵ-AR, then, for 𝑡∈𝐼, the 𝑆-map 𝑓_𝑡 has an extension 𝑆-map 𝐹_𝑡∶(𝑋,•)→(𝑆,∗). Hence 𝐻 has an extension 𝑆-map 𝐹∶(𝑋,•)→𝑃(𝑆,∗) defined by 𝐹(𝑥)(𝑡)=𝐹_𝑡(𝑥),for𝑥∈𝑋,𝑡∈𝐼.(2.3) Hence, 𝑃(𝑆,∗) is an 𝑆_ℵ-AR.

Now in the following definitions, we give the concept of an 𝑆-homotopy extension property in homotopy theory for topological semigroups.

Definition 2.7. Let (𝑆,∗) be a topological semigroup and (𝐴,∗) be an 𝑆-subspace of (𝑆,∗). We will mean by (𝐺^1𝐴,𝐺^2𝑆,𝑂)-maps for   (𝐴,∗)  in (𝑆,∗)  with respect to a topological semigroup (𝑂,∘) the two 𝑆-maps 𝐺^1𝐴∶(𝐴,∗)⟶𝑃(𝑂,∘),𝐺^2𝑆∶(𝑆,∗)⟶(𝑂,∘)(2.4) such that 𝐺^1𝐴(𝑎)(0)=𝐺^2𝑆(𝑎)for𝑎∈𝐴.(2.5)

Definition 2.8. Let (𝑆,∗) and (𝑂,∘) be two topological semigroups. A closed 𝑆-subspace (𝐴,∗) of (𝑆,∗) is said to have 𝑆-homotopy extension property in (𝑆,∗)  with respect to (𝑂,∘) if any (𝐺^1𝐴,𝐺^2𝑆,𝑂)-maps can be extended to an 𝑆-homotopy 𝐹∶(𝑆,∗)→𝑃(𝑂,∘). That is, 𝐹(𝑠)(0)=𝐺^2𝑆(𝑠),𝐹(𝑎)(𝑡)=𝐺^1𝐴(𝑎)(𝑡),for𝑎∈𝐴,𝑠∈𝑆,𝑡∈𝐼.(2.6)

Theorem 2.9. If a closed 𝑆-subspace (𝐴,∗) of (𝑆,∗) has 𝑆-homotopy extension property in (𝑆,∗) with respect to (𝑂,∘), then 𝐴 has homotopy extension property in a space 𝑆 with respect to a space 𝑂.

Proof. Let 𝐺∶(𝑆×{0})∪(𝐴×𝐼)→𝑂 be any continuous map. Define (𝐺^1𝐴,𝐺^2𝑆,𝑂)-maps for (𝐴,𝜋₁) in (𝑆,𝜋₁) with respect to a topological semigroup (𝑂,𝜋₁) by 𝐺^2𝑆(𝑠)=𝐺(𝑠,0) and 𝐺^1𝐴(𝑎)(𝑡)=𝐺(𝑎,𝑡) for all 𝑡∈𝐼, 𝑠𝑥∈𝑆, 𝑎∈𝐴. Then by the hypothesis (𝐺^1𝐴,𝐺^2𝑆,𝑂)-maps can be extended to an 𝑆-homotopy 𝐹∶(𝑆,𝜋₁)→𝑃(𝑂,𝜋₁). That is, 𝐹(𝑠)(0)=𝐺^2𝑆(𝑠),𝐹(𝑎)(𝑡)=𝐺^1𝐴(𝑎)(𝑡),for𝑎∈𝐴,𝑠∈𝑆,𝑡∈𝐼.(2.7) Since 𝐼 is a locally compact and regular space, then by Theorem 1.1 the function 𝐻∶𝑆×𝐼→𝑂 which is defined by 𝐻(𝑠,𝑡)=𝐹(𝑠)(𝑡) for all 𝑡∈𝐼,𝑠∈𝑆 is continuous. Hence 𝐻 is an extension of 𝐺. That is, 𝐴 has homotopy extension property in a space 𝑆 with respect to a space 𝑂.

In the following theorem, we clarify that any closed 𝑆-subspace (𝐴,∗) of a normal topological semigroup (𝑆,∗) has 𝑆-homotopy extension property in (𝑆,∗) with respect to any 𝑆_ℵ-AR space (𝑂,∘).

Theorem 2.10. Let (O,∘) be an 𝑆_ℵ-AR and (𝐴,∗) be a closed 𝑆-subspace of a normal topological semigroup (𝑆,∗). Then (𝐴,∗) has 𝑆-homotopy extension property in (𝑆,∗) with respect to (𝑂,∘).

Proof. Let that there is (𝐺^1𝐴,𝐺^2𝑆,𝑂)-maps. Since 𝐼 is a locally compact and regular, then by Theorem 1.1, for 𝑡∈𝐼−{0}, define an 𝑆-map 𝑓_𝑡∶(𝐴,∗)→(𝑂,∘) by 𝑓_𝑡(𝑎)=𝐺^1𝐴(𝑎)(𝑡),for𝑎∈𝐴.(2.8) Since (𝑂,∘) is an 𝑆_ℵ-AR, then for 𝑡∈𝐼−{0}, the 𝑆-map 𝑓_𝑡 has an extension 𝑆-map 𝐹_𝑡∶(𝑆,∗)→(𝑂,∘). While at the case 𝑡=0, we can take 𝐹₀=𝐺^2𝑆. Hence there is an 𝑆-map 𝐹∶(𝑆,∗)→𝑃(𝑂,∘) defined by 𝐹(𝑠)(𝑡)=𝐺^2𝑆𝐹(𝑠),for𝑠∈𝑆,𝑡=0,_𝑡(𝑠),for𝑠∈𝑆,𝑡∈𝐼−{0}.(2.9) Also we observe that 𝐹(𝑎)(𝑡)=𝐹_𝑡(𝑎)=𝑓_𝑡(𝑎)=𝐺^1𝐴(𝑎)(𝑡),for𝑎∈𝐴.(2.10) That is, (𝐴,∗) has 𝑆-homotopy extension property in (𝑆,∗) with respect to (𝑂,∘).

3. 𝑆-Homotopy Extension Theorems

The general extension theorem for maps of any closed subsets of space 𝑋 into space 𝑌 discusses under which conditions on 𝑋 and 𝑌, for every closed 𝐴⊂𝑋, each map 𝑓∶𝐴→𝑌 will be extendable over 𝑋 relative to 𝑌. The concepts of an AR, an ANR, and homotopy extension property support this theorem in homotopy theory for topological spaces. Now in this section, we introduce the notions of homotopy extension theorem and fiber homotopy extension theorem in homotopy theory for topological semigroups via 𝑆_ℵ-AR property.

Lemma 3.1. Let (𝑆,∗) be a topological semigroup. For each path 𝛼 in (𝑆,∗) and 𝑟∈𝐼, let 𝛼_𝑟 be a path in (𝑆,∗) defined by 𝛼_𝑟(𝑡)=𝛼[ℎ(𝑟,𝑡)] for all 𝑡∈𝐼, where ℎ∶𝐼×𝐼→𝐼 be any map. Then the function 𝐻∶𝑃(𝑆,∗)→𝑃(𝑃𝑎(𝑆),𝑝(∗)) defined by 𝐻(𝛼)(𝑟)=𝛼_𝑟for𝑟∈𝐼,𝛼∈𝑃𝑎(𝑆)(3.1) is an 𝑆-map.

Proof. By Theorem 1.1 to prove that 𝐻 is continuous, it is sufficient to prove that the function 𝐹∶𝑃𝑎(𝑆)×𝐼→𝑃𝑎(𝑆) defined by 𝐹(𝛼,𝑟)=𝛼_𝑟 for all 𝑟∈𝐼, 𝛼∈𝑃𝑎(𝑆) is continuous. Let (𝛽,𝑟_𝑜)∈𝑃𝑎(𝑆)×𝐼 and 𝑊(𝐾,𝑈) be a neighborhood of 𝛽_𝑟𝑜 in 𝑃𝑎(𝑆). Then 𝛽(ℎ({𝑟_𝑜}×𝐾))=𝛽_𝑟𝑜(𝐾)⊆𝑈. By the continuity of 𝛽 and 𝑈 is an open set containing 𝛽(ℎ({𝑟_𝑜}×𝐾)), there is open set 𝐺 in 𝐼 such that ℎ({𝑟_𝑜}×𝐾)⊆𝐺 and 𝛽(𝐺)⊆𝑈. Also by the continuity of ℎ, there are two open sets 𝐼_𝑟𝑜 and 𝐼_𝐾 in 𝐼 such that 𝑟_𝑜×𝐾⊆𝐼_𝑟𝑜×𝐼_𝐾,ℎ𝐼_𝑟𝑜×𝐼_𝐾⊆𝐺.(3.2) Since 𝐼_𝑟𝑜 is an open set in 𝐼 containing 𝑟_𝑜, then there is a positive number 𝜖>0 such that 𝑟_𝑜∈𝑟_𝑜−𝜖3,𝑟^𝑜+𝜖3⊂𝐷=𝑟^𝑜−𝜖2,𝑟^𝑜+𝜖2⊂𝑟^𝑜−𝜖,𝑟_𝑜+𝜖⊆𝐼_𝑟𝑜.(3.3)Since 𝐷 and 𝐾 are compact sets in 𝐼 and ℎ is a continuous, then ℎ(𝐷×𝐾) is also a compact set in 𝐼. Now consider that 𝑊(ℎ(𝐷×𝐾),𝑈)×(𝑟_𝑜−𝜖/3,𝑟_𝑜+𝜖/3) a neighborhood of (𝛽,𝑟_𝑜) in 𝑃𝑎(𝑆)×𝐼. Hence for (𝛼,𝑟)∈𝑊(ℎ(𝐷×𝐾),𝑈)×𝑟_𝑜−𝜖3,𝑟^𝑜+𝜖𝐹(𝛼,𝑟)(𝐾)=𝛼_𝑟3,(𝐾)=𝛼(ℎ({𝑟}×𝐾))⊆𝛼[ℎ(𝐷×𝐾)]⊆𝑈.(3.4) That is, 𝐹(𝛼,𝑟)∈𝑊(𝐾,𝑈). Hence the function 𝐹 is continuous.
Now for 𝛼,𝛽∈𝑃(𝑆,∗), {𝐻[𝑝(∗)(𝛼,𝛽)](𝑟)}(𝑡)====∗∗[[𝑝𝑝[((𝛼𝛼∗∗_𝑟())ℎ((𝛼𝛼(𝑟,,,𝛽𝛽𝑡)))]])_𝑟(,(ℎ𝛽𝑡)((𝑟ℎ,𝑡()𝑟),𝑡))]=𝑝(∗)(𝑡)𝛼,_𝑟𝛽_𝑟(𝑡)={𝑝(𝑝(∗)),[𝛽𝐻_𝑟(𝛼()𝑡),𝐻(𝛽)](𝑟)}(𝑡).(3.5) Hence 𝐻[𝑝(∗)(𝛼,𝛽)]=𝑝(𝑝(∗))[𝐻(𝛼),𝐻(𝛽)], that is, 𝐻 is an 𝑆-map.

The following theorem clarifies the notion of 𝑆-fiber homotopy extension theorem in homotopy theory for topological semigroups via 𝑆_ℵ-AR property

Theorem 3.2. Let 𝑓∶(𝑆,∗)→(𝑂,∘) be an 𝑆_ℵ-regular fibration with an 𝑆_ℵ-AR (𝑆,∗). Let (𝐴,•) be a closed 𝑆-subspace of a normal topological semigroup (𝑋,•). If there is (𝐺^1𝐴,𝐺^2𝑋,𝑆)-maps such that 𝑓𝐺^1𝐴(𝑎)(𝑡)=𝑓𝐺^1𝐴(𝑎)(0)for𝑎∈𝐴,𝑡∈𝐼,(3.6) then there is an 𝑆-map 𝐻∶(𝑋,•)→𝑃(𝑆,∗) such that 𝐻 is an extension of 𝐺^1𝐴, 𝐻₀=𝐺^2𝑋 and 𝑓[𝐻(𝑥)(𝑡)]=𝑓[𝐻(𝑥)(0)]for𝑥∈𝑋,𝑡∈𝐼.(3.7)

Proof. Since (𝑆,∗) is an 𝑆_ℵ-AR and (𝐴,•) is a closed 𝑆-subspace of a normal topological semigroup (𝑋,•), then, by Theorem 2.10, the 𝑆-map 𝐺^1𝐴 can be extended to 𝑆-map 𝐺∶(𝑋,•)→𝑃(𝑆,∗) such that 𝐺₀=𝐺^2𝑋. Now for 𝛼∈𝑃(𝑆,∗) and 𝑟∈𝐼, we can define the path 𝛼_𝑟 in 𝑃(𝑆,∗) by 𝛼_𝑟(𝑡)=𝛼[(1−𝑡)𝑟] for all 𝑡∈𝐼. Then by Lemma 3.1, the function 𝑃(𝑆,∗)⟶𝑃(𝑃𝑎(𝑆),𝑝(∗))∶(𝛼)(𝑟)⟶𝛼_𝑟(3.8) is an 𝑆-map. Hence, we can define the 𝑆-map 𝐻∶(𝑋,•)→𝑃(𝑆,∗) by 𝐻(𝑥)(𝑡)=𝐿_𝑓𝐺(𝑥)(𝑡),𝑓∘𝐺(𝑥)_𝑡(1)for𝑥∈𝑋,𝑡∈𝐼.(3.9) Firstly, we show that 𝐻₀=𝐺^2𝑋. By the 𝑆_ℵ-regularity of 𝐿_𝑓 and since 𝐺 is an extension for 𝐺^1𝐴, we observe that for 𝑥∈𝑋, 𝐻(𝑥)(0)==𝐿𝐿_𝑓𝑓𝐺(𝑥)(0),𝑓∘𝐺(𝑥)₀(1)=𝐿_𝑓(𝐺(𝑥)(0),𝑓∘𝐺(𝑥)(0))(1)=𝐺(𝑥𝐺)((0𝑥))((01)),=𝑓∘𝐺𝐺(𝑥(𝑥))(0(0))=(𝐺1_2𝑋)(𝑥).(3.10) Secondly, we show that 𝐻 is an extension of 𝐺^1𝐴. Since 𝐺 is an extension for 𝐺^1𝐴 and by the hypothesis, we get that 𝑓∘𝐺(𝑎)_𝑟(𝑡)==𝑓𝑓[𝐺𝐺_1𝐴(𝑎)((1−𝑡)𝑟)]=𝑓𝐺_1𝐴(𝑎)((1−𝑡)𝑟)=𝑓𝐺_1𝐴(𝑎)(0)=𝑓[𝐺(𝑎()𝑎()𝑟()𝑟])=𝑓∘𝐺(𝑎)(𝑟)(𝑡),(3.11) for all 𝑎∈𝐴 and 𝑟,𝑡∈𝐼. Hence, by the 𝑆_ℵ-regularity of 𝐿_𝑓, we get that 𝐻(𝑎)(𝑟)=𝐺(𝑎)(𝑟)=𝐺^1𝐴(𝑎)(𝑟)for𝑟∈𝐼,𝑎∈𝐴.(3.12) Hence 𝐻 is an extension for 𝐺^1𝐴. Finally, we also observe that 𝑓[𝐻(𝑥)(𝑡)]==𝑓𝑓𝐿∘𝐺_𝑓(𝑥𝐺)_𝑡(𝑥)(𝑡),𝑓∘𝐺(𝑥)_𝑡(1)=𝑓𝐺(𝑥)_𝑡(1)==𝑓𝑓[𝐺𝐺_2𝑋(𝑥)((01))](𝑥)=𝑓[𝐻(𝑥)(0)],(3.13) for all 𝑥∈𝑋, 𝑡∈𝐼.

Now we can give the other rephrasing of above theorem in the following corollary.

Corollary 3.3. Let 𝑓∶(𝑆,∗)→(𝑂,∘) be an 𝑆_ℵ-regular fibration with an 𝑆_ℵ-AR (𝑆,∗) and (𝐴,•) be a closed 𝑆-subspace of a normal topological semigroup (𝑋,•). Let 𝑘₁,𝑘₂∶(𝐴,•)→(𝑆,∗) be two 𝑆-maps and 𝑅∶(𝐴,•)→𝑃(𝑆,∗) be an 𝑆-homotopy between them such that 𝑓[𝑅(𝑎)(𝑡)]=𝑓[𝑅(𝑎)(0)]for𝑎∈𝐴,𝑡∈𝐼.(3.14) If 𝑘₁ has an extension 𝑆-map 𝐾₁ to all of (𝑋,•), then 𝑘₂ has an extension 𝑆-map 𝐾₂ to all of (𝑋,•). Also there is an 𝑆-homotopy 𝐻∶(𝑋,•)→𝑃(𝑆,∗) between 𝐾₁ and 𝐾₂ such that 𝐻 is an extension of 𝑅 and 𝑓[𝐻(𝑥)(𝑡)]=𝑓[𝐻(𝑥)(0)]for𝑥∈𝑋,𝑡∈𝐼.(3.15)

Proof. Since 𝑘₁ has an extension 𝑆-map 𝐾₁ to all of (𝑋,•), that is, 𝐾₁(𝑎)=𝑘₁(𝑎)=𝑅(𝑎)(0)for𝑎∈𝐴.(3.16) Hence there is (𝑅,𝐾₁,𝑆)-maps with the property 𝑓[𝑅(𝑎)(𝑡)]=𝑓[𝑅(𝑎)(0)]for𝑎∈𝐴,𝑡∈𝐼.(3.17) Then by Theorem 3.2, there is an 𝑆-map 𝐻∶(𝑋,•)→𝑃(𝑆,∗) such that 𝐻 is an extension of 𝑅, 𝐻₀=𝐾₁ and 𝑓[𝐻(𝑥)(𝑡)]=𝑓[𝐻(𝑥)(0)]for𝑥∈𝑋,𝑡∈𝐼.(3.18) Also we can define the 𝑆-map 𝐾₂∶(𝑋,•)→(𝑆,∗) by 𝐾₂(𝑥)=𝐻(𝑥)(1) for all 𝑥∈𝑋. Then 𝐻 is an 𝑆-homotopy between 𝐾₁ and 𝐾₂ and also 𝐾₂(𝑎)=𝐻(𝑎)(1)=𝑘₂(𝑎)for𝑎∈𝐴.(3.19) That is, 𝐾₂ is an extension of 𝑘₂.

The following theorem clarifies the notion of 𝑆-fiber homotopy extension theorem via 𝑆_ℵ-AR property.

Theorem 3.4. Let 𝑓₁∶(𝑆₁,∗₁)→(𝑂,∘) and 𝑓₂∶(𝑆₂,∗₂)→(𝑂,∘) be two 𝑆_ℵ-fibrations with an 𝑆_ℵ-AR (𝑆₂,∗₂), a normal (𝑆₁,∗₁) and (𝐸,∘) be a closed 𝑆-subspace of (𝑂,∘). Let 𝑘,𝑘^′∶(𝑓¹⁻¹(𝐸),∗₁)→(𝑓²⁻¹(𝐸),∗₂) be two 𝑆-fiber maps between two 𝑆_ℵ-fibrations 𝑓₁|𝐸 and 𝑓₂|𝐸 such that 𝐾−^𝑓_≈𝐾^. If 𝑘 is extendable to a full 𝑆-fiber map 𝐾∶(𝑆₁,∗₁)→(𝑆₂,∗₂), then 𝑘^ is extendable to a full 𝑆-fiber map 𝐾^∶(𝑆₁,∗₁)→(𝑆₂,∗₂) and 𝐾−^𝑓_≈𝐾^.

Proof. Since 𝐾−^𝑓_≈𝐾^, then there is an 𝑆−homotopy 𝑅∶𝑓¹⁻¹(𝐸),∗₁⟶𝑃𝑓²⁻¹(𝐸),∗₂⊆𝑃𝑆₂,∗₂,(3.20) such that 𝑅(𝑠)(0)=𝑘(𝑠), 𝑅(𝑠)(1)=𝑘^(𝑠) and 𝑓₂[𝑅(𝑠)(𝑡)]=𝑓₁(𝑠)=𝑓₂[𝑘(𝑠)]=𝑓₂[𝑅(𝑠)(0)]for𝑠∈𝑓¹⁻¹(𝐸),𝑡∈𝐼.(3.21) Since 𝑘 is extendable to a full 𝑆-fiber map 𝐾, then there is (𝐾,𝑅,𝑆₂)-maps. Since (𝑓¹⁻¹(𝐸),∗₁) is a closed 𝑆-subspace of a normal (𝑆₁,∗₁), we can apply Theorem 3.2 on 𝑆_ℵ-fibration 𝑓₂, where 𝐴=𝑓¹⁻¹(𝐸) and 𝑋=(𝑆₁,∗₁). Hence there is an 𝑆-map 𝐻∶(𝑆₁,∗₁)→𝑃(𝑆₂,∗₂) such that 𝐻 is an extension of 𝑅, 𝐻₀=𝐾 and 𝑓₂[𝐻(𝑠)(𝑡)]=𝑓₂[𝐻(𝑠)(0)]for𝑠∈𝑆₁,𝑡∈𝐼.(3.22) Now we can define 𝐾^′ by 𝐾^=𝐻₁. Hence 𝐻 is an 𝑆-homotopy between 𝐾 and 𝐾^. From (3.22) and since 𝐾 is an 𝑆-fiber map, we get that 𝐾^ is an 𝑆-fiber map and 𝑓₂[𝐻(𝑠)(𝑡)]=𝑓₂[𝐻(𝑠)(0)]=𝑓₂[𝐾(𝑠)]=𝑓₁(𝑠)for𝑠∈𝑆₁,𝑡∈𝐼.(3.23) Hence𝐾−^𝑓_≈𝐾^′.