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ISRN Geometry
Volume 2012 (2012), Article ID 359403, 9 pages
http://dx.doi.org/10.5402/2012/359403
Research Article

Homotopy Extension Property in Homotopy Theory for Topological Semigroups

Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia

Received 2 November 2011; Accepted 13 December 2011

Academic Editor: D. Danielli

Copyright © 2012 Adem Kılıçman and Amin Saif. 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

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-connectedness, 𝑆-homotopy domination, 𝑆-contractibility, and 𝑆-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,𝑦𝐴. The pair (𝑆𝐼,𝑝()) is a topological semigroup with the compact-open topology on a path space 𝑆𝐼, where the multiplication 𝑝()𝑆𝐼×𝑆𝐼𝑆𝐼 on 𝑃𝑎(𝑆) defined by[]𝑝()(𝛼,𝛽)(𝑡)=(𝛼(𝑡),𝛽(𝑡)),for𝛼,𝛽𝑆𝐼,𝑡𝐼.(1.1)

𝑃(𝑆,) will denote a topological semigroup (𝑆𝐼,𝑝()). For 𝑠𝑆, ̃𝑠 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,𝑦𝑆1.(1.2)

Two 𝑆-maps 𝑓,𝑔(𝑆,)(𝑂,) are called 𝑆-homotopic (written 𝑓𝑠𝑔) 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 𝑠𝑆, for more details see [13].

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

where Δ𝑓={(𝑠,𝛼)𝑆×𝑃𝑎(𝑂)𝑓(𝑠)=𝛼(0)}. If 𝐿𝑓(𝑠,𝑓̃𝑠)=̃𝑠 for all 𝑠𝑆, then the 𝑆-lifting function is called an 𝑆-regular lifting function. Also we say that the 𝑆-map 𝑓(𝑆,)(𝑂,) is an 𝑆-regular fibration if it has 𝑆-regular lifting function.

Note that we can easily prove that 𝑆-map 𝑓 is an 𝑆-regular fibration if and only if it has 𝑆-regular lifting function. Further, if we let 𝑓(𝑆,)(𝑂,) be an 𝑆-fibration and (𝐸,) be an 𝑆-subspace of (𝑂,). Then 𝑆-map 𝑓|𝑓1(𝐸)(𝑓1(𝐸),)(𝐸,) is an 𝑆-fibration, where 𝑓|𝐸 will be the 𝑆-fibration 𝑓|𝑓1(𝐸). Thus, one by defining two 𝑆-fibrations 𝑓1(𝑆1,1)(𝑂,) and 𝑓2(𝑆2,2)(𝑂,), an 𝑆-map (𝑆1,1)(𝑆2,2) and setting the relation 𝑓2=𝑓1, then we can also study two 𝑆-fiber maps (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 𝐹𝑋×𝑌𝑍 always gives rise to a map 𝐻𝑋𝐶(𝑌,𝑍) by defining 𝐻(𝑥)(𝑦)=𝐹(𝑥,𝑦) for all 𝑥𝑋,𝑦𝑌. If 𝑌 is locally compact and regular space, then the map 𝐻𝑋𝐶(𝑌,𝑍) always gives rise to map 𝐹𝑋×𝑌𝑍 by defining 𝐹(𝑥,𝑦)=𝐻(𝑥)(𝑦) for all 𝑥𝑋,𝑦𝑌.

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

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 𝑠𝑂. 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-absolute retract (𝑆-AR) for normal topological semigroups (𝑋,) in the class 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-absolute neighborhood retract (𝑆-ANR) for normal topological semigroups (𝑋,) in the class 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-AR (resp. 𝑆-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 𝑥𝑀 and 𝑥𝐽, that is, 1/2<𝑥1. Since 𝑥/2𝐽 and an 𝑆-map 𝑅 would have to satisfy an impossible condition:𝑥2𝑥=𝑟21=𝑅𝑥21=𝑅(𝑥)𝑅2=12𝑅(𝑥)(2.1)

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

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

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

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

Theorem 2.6. If(𝑆,)is an𝑆-AR, then𝑃(𝑆,)is also an𝑆-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 𝑡𝐼, we can define an 𝑆-map 𝑓𝑡(𝐴,)(𝑆,) by 𝑓𝑡(𝑎)=𝐻(𝑎)(𝑡),for𝑎𝐴.(2.2) Since (𝑆,) is an 𝑆-AR, then, for 𝑡𝐼, the 𝑆-map 𝑓𝑡 has an extension 𝑆-map 𝐹𝑡(𝑋,)(𝑆,). Hence 𝐻 has an extension 𝑆-map 𝐹(𝑋,)𝑃(𝑆,) defined by 𝐹(𝑥)(𝑡)=𝐹𝑡(𝑥),for𝑥𝑋,𝑡𝐼.(2.3) Hence, 𝑃(𝑆,) is an 𝑆-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 (𝐺1𝐴,𝐺2𝑆,𝑂)-maps for   (𝐴,)  in (𝑆,)  with respect to a topological semigroup (𝑂,) the two 𝑆-maps 𝐺1𝐴(𝐴,)𝑃(𝑂,),𝐺2𝑆(𝑆,)(𝑂,)(2.4) such that 𝐺1𝐴(𝑎)(0)=𝐺2𝑆(𝑎)for𝑎𝐴.(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 (𝐺1𝐴,𝐺2𝑆,𝑂)-maps can be extended to an 𝑆-homotopy 𝐹(𝑆,)𝑃(𝑂,). That is, 𝐹(𝑠)(0)=𝐺2𝑆(𝑠),𝐹(𝑎)(𝑡)=𝐺1𝐴(𝑎)(𝑡),for𝑎𝐴,𝑠𝑆,𝑡𝐼.(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 (𝐺1𝐴,𝐺2𝑆,𝑂)-maps for (𝐴,𝜋1) in (𝑆,𝜋1) with respect to a topological semigroup (𝑂,𝜋1) by 𝐺2𝑆(𝑠)=𝐺(𝑠,0) and 𝐺1𝐴(𝑎)(𝑡)=𝐺(𝑎,𝑡) for all 𝑡𝐼, 𝑠𝑥𝑆, 𝑎𝐴. Then by the hypothesis (𝐺1𝐴,𝐺2𝑆,𝑂)-maps can be extended to an 𝑆-homotopy 𝐹(𝑆,𝜋1)𝑃(𝑂,𝜋1). That is, 𝐹(𝑠)(0)=𝐺2𝑆(𝑠),𝐹(𝑎)(𝑡)=𝐺1𝐴(𝑎)(𝑡),for𝑎𝐴,𝑠𝑆,𝑡𝐼.(2.7) Since 𝐼 is a locally compact and regular space, then by Theorem 1.1 the function 𝐻𝑆×𝐼𝑂 which is defined by 𝐻(𝑠,𝑡)=𝐹(𝑠)(𝑡) for all 𝑡𝐼,𝑠𝑆 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-AR space (𝑂,).

Theorem 2.10. Let (O,) be an 𝑆-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 (𝐺1𝐴,𝐺2𝑆,𝑂)-maps. Since 𝐼 is a locally compact and regular, then by Theorem 1.1, for 𝑡𝐼{0}, define an 𝑆-map 𝑓𝑡(𝐴,)(𝑂,) by 𝑓𝑡(𝑎)=𝐺1𝐴(𝑎)(𝑡),for𝑎𝐴.(2.8) Since (𝑂,) is an 𝑆-AR, then for 𝑡𝐼{0}, the 𝑆-map 𝑓𝑡 has an extension 𝑆-map 𝐹𝑡(𝑆,)(𝑂,). While at the case 𝑡=0, we can take 𝐹0=𝐺2𝑆. Hence there is an 𝑆-map 𝐹(𝑆,)𝑃(𝑂,) defined by 𝐺𝐹(𝑠)(𝑡)=2𝑆𝐹(𝑠),for𝑠𝑆,𝑡=0,𝑡(𝑠),for𝑠𝑆,𝑡𝐼{0}.(2.9) Also we observe that 𝐹(𝑎)(𝑡)=𝐹𝑡(𝑎)=𝑓𝑡(𝑎)=𝐺1𝐴(𝑎)(𝑡),for𝑎𝐴.(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 𝐴𝑋, each map 𝑓𝐴𝑌 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-AR property.

Lemma 3.1. Let (𝑆,) be a topological semigroup. For each path 𝛼 in (𝑆,) and 𝑟𝐼, let 𝛼𝑟 be a path in (𝑆,) defined by 𝛼𝑟(𝑡)=𝛼[(𝑟,𝑡)] for all 𝑡𝐼, where 𝐼×𝐼𝐼 be any map. Then the function 𝐻𝑃(𝑆,)𝑃(𝑃𝑎(𝑆),𝑝()) defined by 𝐻(𝛼)(𝑟)=𝛼𝑟for𝑟𝐼,𝛼𝑃𝑎(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 𝐹𝑃𝑎(𝑆)×𝐼𝑃𝑎(𝑆) defined by 𝐹(𝛼,𝑟)=𝛼𝑟 for all 𝑟𝐼, 𝛼𝑃𝑎(𝑆) is continuous. Let (𝛽,𝑟𝑜)𝑃𝑎(𝑆)×𝐼 and 𝑊(𝐾,𝑈) be a neighborhood of 𝛽𝑟𝑜 in 𝑃𝑎(𝑆). Then 𝛽(({𝑟𝑜}×𝐾))=𝛽𝑟𝑜(𝐾)𝑈. By the continuity of 𝛽 and 𝑈 is an open set containing 𝛽(({𝑟𝑜}×𝐾)), there is open set 𝐺 in 𝐼 such that ({𝑟𝑜}×𝐾)𝐺 and 𝛽(𝐺)𝑈. Also by the continuity of , there are two open sets 𝐼𝑟𝑜 and 𝐼𝐾 in 𝐼 such that 𝑟𝑜×𝐾𝐼𝑟𝑜×𝐼𝐾𝐼,𝑟𝑜×𝐼𝐾𝐺.(3.2) Since 𝐼𝑟𝑜 is an open set in 𝐼 containing 𝑟𝑜, then there is a positive number 𝜖>0 such that 𝑟𝑜𝑟𝑜𝜖3,𝑟𝑜+𝜖3𝑟𝐷=𝑜𝜖2,𝑟𝑜+𝜖2𝑟𝑜𝜖,𝑟𝑜+𝜖𝐼𝑟𝑜.(3.3)Since 𝐷 and 𝐾 are compact sets in 𝐼 and is a continuous, then (𝐷×𝐾) is also a compact set in 𝐼. Now consider that 𝑊((𝐷×𝐾),𝑈)×(𝑟𝑜𝜖/3,𝑟𝑜+𝜖/3) a neighborhood of (𝛽,𝑟𝑜) in 𝑃𝑎(𝑆)×𝐼. Hence for 𝑟(𝛼,𝑟)𝑊((𝐷×𝐾),𝑈)×𝑜𝜖3,𝑟𝑜+𝜖3,𝐹(𝛼,𝑟)(𝐾)=𝛼𝑟[](𝐾)=𝛼(({𝑟}×𝐾))𝛼(𝐷×𝐾)𝑈.(3.4) That is, 𝐹(𝛼,𝑟)𝑊(𝐾,𝑈). Hence the function 𝐹 is continuous.
Now for 𝛼,𝛽𝑃(𝑆,), [][]{𝐻𝑝()(𝛼,𝛽)(𝑟)}(𝑡)=𝑝()(𝛼,𝛽)𝑟=[𝑝][]𝛼(𝑡)()(𝛼,𝛽)((𝑟,𝑡))=𝛼((𝑟,𝑡)),𝛽((𝑟,𝑡))=𝑟(𝑡),𝛽𝑟(=𝑝𝛼𝑡)()𝑟,𝛽𝑟[](𝑡)={𝑝(𝑝())𝐻(𝛼),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-AR property

Theorem 3.2. Let 𝑓(𝑆,)(𝑂,) be an 𝑆-regular fibration with an 𝑆-AR (𝑆,). Let (𝐴,) be a closed 𝑆-subspace of a normal topological semigroup (𝑋,). If there is (𝐺1𝐴,𝐺2𝑋,𝑆)-maps such that 𝑓𝐺1𝐴𝐺(𝑎)(𝑡)=𝑓1𝐴(𝑎)(0)for𝑎𝐴,𝑡𝐼,(3.6) then there is an 𝑆-map 𝐻(𝑋,)𝑃(𝑆,) such that 𝐻 is an extension of 𝐺1𝐴, 𝐻0=𝐺2𝑋 and 𝑓[][]𝐻(𝑥)(𝑡)=𝑓𝐻(𝑥)(0)for𝑥𝑋,𝑡𝐼.(3.7)

Proof. Since (𝑆,) is an 𝑆-AR and (𝐴,) is a closed 𝑆-subspace of a normal topological semigroup (𝑋,), then, by Theorem 2.10, the 𝑆-map 𝐺1𝐴 can be extended to 𝑆-map 𝐺(𝑋,)𝑃(𝑆,) such that 𝐺0=𝐺2𝑋. Now for 𝛼𝑃(𝑆,) and 𝑟𝐼, we can define the path 𝛼𝑟 in 𝑃(𝑆,) by 𝛼𝑟(𝑡)=𝛼[(1𝑡)𝑟] for all 𝑡𝐼. Then by Lemma 3.1, the function 𝑃(𝑆,)𝑃(𝑃𝑎(𝑆),𝑝())(𝛼)(𝑟)𝛼𝑟(3.8) is an 𝑆-map. Hence, we can define the 𝑆-map 𝐻(𝑋,)𝑃(𝑆,) by 𝐻(𝑥)(𝑡)=𝐿𝑓𝐺(𝑥)(𝑡),𝑓𝐺(𝑥)𝑡(1)for𝑥𝑋,𝑡𝐼.(3.9) Firstly, we show that 𝐻0=𝐺2𝑋. By the 𝑆-regularity of 𝐿𝑓 and since 𝐺 is an extension for 𝐺1𝐴, we observe that for 𝑥𝑋, 𝐻(𝑥)(0)=𝐿𝑓𝐺(𝑥)(0),𝑓𝐺(𝑥)0(1)=𝐿𝑓(𝐺(𝑥)(0),𝑓𝐺(𝑥)(0))(1)=𝐿𝑓=𝐺(𝑥)(0),𝑓𝐺(𝑥)(0)(1)𝐺(𝑥)(0)(1)=𝐺(𝑥)(0)=𝐺2𝑋(𝑥).(3.10) Secondly, we show that 𝐻 is an extension of 𝐺1𝐴. Since 𝐺 is an extension for 𝐺1𝐴 and by the hypothesis, we get that 𝑓𝐺(𝑎)𝑟[𝐺]𝐺(𝑡)=𝑓(𝑎)((1𝑡)𝑟)=𝑓1𝐴𝐺(𝑎)((1𝑡)𝑟)=𝑓1𝐴𝐺(𝑎)(0)=𝑓1𝐴[]=(𝑎)(𝑟)=𝑓𝐺(𝑎)(𝑟)𝑓𝐺(𝑎)(𝑟)(𝑡),(3.11) for all 𝑎𝐴 and 𝑟,𝑡𝐼. Hence, by the 𝑆-regularity of 𝐿𝑓, we get that 𝐻(𝑎)(𝑟)=𝐺(𝑎)(𝑟)=𝐺1𝐴(𝑎)(𝑟)for𝑟𝐼,𝑎𝐴.(3.12) Hence 𝐻 is an extension for 𝐺1𝐴. Finally, we also observe that 𝑓[𝐻]𝐿(𝑥)(𝑡)=𝑓𝑓𝐺(𝑥)(𝑡),𝑓𝐺(𝑥)𝑡=(1)𝑓𝐺(𝑥)𝑡(1)=𝑓𝐺(𝑥)𝑡[]𝐺(1)=𝑓𝐺(𝑥)(0)=𝑓2𝑋([],𝑥)=𝑓𝐻(𝑥)(0)(3.13) for all 𝑥𝑋, 𝑡𝐼.

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

Corollary 3.3. Let 𝑓(𝑆,)(𝑂,) be an 𝑆-regular fibration with an 𝑆-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)=𝑓𝑅(𝑎)(0)for𝑎𝐴,𝑡𝐼.(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)=𝑓𝐻(𝑥)(0)for𝑥𝑋,𝑡𝐼.(3.15)

Proof. Since 𝑘1 has an extension 𝑆-map 𝐾1 to all of (𝑋,), that is, 𝐾1(𝑎)=𝑘1(𝑎)=𝑅(𝑎)(0)for𝑎𝐴.(3.16) Hence there is (𝑅,𝐾1,𝑆)-maps with the property 𝑓[][]𝑅(𝑎)(𝑡)=𝑓𝑅(𝑎)(0)for𝑎𝐴,𝑡𝐼.(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)=𝑓𝐻(𝑥)(0)for𝑥𝑋,𝑡𝐼.(3.18) Also we can define the 𝑆-map 𝐾2(𝑋,)(𝑆,) by 𝐾2(𝑥)=𝐻(𝑥)(1) for all 𝑥𝑋. Then 𝐻 is an 𝑆-homotopy between 𝐾1 and 𝐾2 and also 𝐾2(𝑎)=𝐻(𝑎)(1)=𝑘2(𝑎)for𝑎𝐴.(3.19) That is, 𝐾2 is an extension of 𝑘2.

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

Theorem 3.4. Let 𝑓1(𝑆1,1)(𝑂,) and 𝑓2(𝑆2,2)(𝑂,) be two 𝑆-fibrations with an 𝑆-AR (𝑆2,2), a normal (𝑆1,1) and (𝐸,) be a closed 𝑆-subspace of (𝑂,). Let 𝑘,𝑘(𝑓11(𝐸),1)(𝑓21(𝐸),2) be two 𝑆-fiber maps between two 𝑆-fibrations 𝑓1|𝐸 and 𝑓2|𝐸 such that 𝐾𝑓𝐾. 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 𝐾𝑓𝐾.

Proof. Since 𝐾𝑓𝐾, then there is an 𝑆homotopy 𝑓𝑅11(𝐸),1𝑓𝑃21(𝐸),2𝑆𝑃2,2,(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)for𝑠𝑓11(𝐸),𝑡𝐼.(3.21) Since 𝑘 is extendable to a full 𝑆-fiber map 𝐾, then there is (𝐾,𝑅,𝑆2)-maps. Since (𝑓11(𝐸),1) is a closed 𝑆-subspace of a normal (𝑆1,1), we can apply Theorem 3.2 on 𝑆-fibration 𝑓2, where 𝐴=𝑓11(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)for𝑠𝑆1,𝑡𝐼.(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)for𝑠𝑆1,𝑡𝐼.(3.23) Hence𝐾𝑓𝐾.

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