# OnCℵ-Fibrations in Bitopological Semigroups

Authors: Suliman Dawood; Adem Kılıçman
Journal: The Scientific World Journal
 (2014)

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## Abstract

We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in theCℵ-fibration property. We give and prove the relationship between the Cℵ-fibration property and an approximate fibration property. Furthermore, we study the pullback maps for Cℵ-fibrations.


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## Body

## 1. Introduction

In homotopy theory for topological space (i.e., spaces), Hurewicz [1] introduced the concepts of fibrations and path lifting property of maps and showed its equivalence with the covering homotopy property. Coram and Duvall [2] introduced approximate fibrations as a generalization of cell-like maps [3] and showed that the uniform limit of a sequence of Hurewicz fibrations is an approximate fibration. In 1963, Kelly [4] introduced the notion of bitopological spaces. Such spaces were equipped with its two (arbitrary) topologies. The reader is suggested to refer to [4] for the detail definitions and notations. The concept of homotopy theory for topological semigroups has been introduced by Cerin in 2002 [5]. In this theory, he introduced Sℵ-fibrations as extension of Hurewicz fibrations. In [6], we introduced the concepts of bitopological semigroups, c-bitopological semigroups, and Cℵ-fibrations as extension of Sℵ-fibrations.This paper is organized as follows. It consists of five sections. After this Introduction, Section2 is devoted to some preliminaries. In Section 3 we show the pullbacks of S-maps which have the Cℵ-fibration property that will also have this property and the pullbacks of Cℵ-fibrations are Cℵ-fibrations under given conditions. In Section 4 we develop and extend path lifting property in homotopy theory for topological semigroups to theory for bitopological semigroups. Some results about Hurewicz fibrations carry over. In Section 5 we give and prove the relationship between the CNπ-fibration property and an approximate fibration property.


## 2. Preliminaries

Throughout this paper, by allXτ we mean all topological spaces (X,τ) which will be assumed Hausdorff spaces. By all Xτ12 we mean all bitopological spaces (X,τ1,τ2). For two bitopological spaces Xτ12 and Yρ12, ap-maph:Xτ12→Yρ12 is a function from X into Y that is continuous function (i.e., a map) from a space Xτ1 into a space Yρ1 and from Xτ2 into Yρ2 [4].Recall [5] that atopological semigroup or anS-space is a pair (Xτ,*) consisting of a topological space Xτ and a map *:Xτ×Xτ→Xτ from the product space Xτ×Xτ into Xτ such that *(x,*(y,z))=*(*(x,y),z) for all x,y,z∈X. AnS-space (A,*′) is called anS-subspace of (Xτ,*) if A is a subspace of Xτ and the map * takes the product A×A into A and *′(x,y)=*(x,y) for all x,y∈A. We denote the class of allS-spaces by ℵ. For every space Xτ, by P(Xτ), we mean the space of all paths from the unit closed interval I=[0,1] into Xτ with the compact-open topology. Recall [5] that, for everyS-space (Xτ,*), (P(Xτ),p(*)) is anS-space where p(*):P(Xτ)×P(Xτ)→P(Xτ) is a map defined by p(*)(α,β)(t)=*(α(t),β(t)) for all α,β∈P(Xτ), t∈I. The shorter notion for thisS-space will be P(Xτ,*). For every space Xτ, thenatural S-space is anS-space (Xτ,πi), where πi is a continuous associative multiplication on Xτ given by π1(x,y)=x and π2(x,y)=y for all x,y∈X. We denote the class of all naturalS-spaces (Xτ,π) by Nπ, where π=π1,π2.Recall [5] that the function f:(Xτ,*)→(Oρ,∘) is called anS-map if f is a map of a space Xτ into Oρ and f(*(x,y))=∘(f(x),f(y)) for all x,y∈X. The function f:Xτ→Oρ of a naturalS-space (Xτ,π) into (Oρ,π) is anS-map if and only if it is continuous. TheS-maps f,g:(Xτ,*)→(Oρ,∘) are calledS-homotopic and write f≃sg provided there is anS-map H:(Xτ,*)→P(Oρ,∘) called anS-homotopy such that H(x)(0)=f(x) and H(x)(1)=g(x) for all x∈X.Abitopological semigroup is a pair (Xτ12,*) consisting of a bitopological space Xτ12 and the associative multiplication * on X such that * is an p-map from the product bitopological space (X×X,τ1×τ1,τ2×τ2) into Xτ12. For B⊆X, by B|τ12 we mean the bitopological subspace (B,τ1|B,τ2|B) of Xτ12. If the p-map * takes the product B×B into B then the pair (B|τ12,*) will be a bitopological semigroup and will be called anb-subspace of (Xτ12,*).The functionh:(Xτ12,*)→(Yρ12,∘) is called an Si-map from (Xτ12,*) into (Yρ12,∘) provided h is anS-map from a functionS-space (Xτi,*) into anS-space (Yρi,∘), where i=1,2. We say that h is anSp-map if it is an S1-map and S2-map.Anc-bitopological semigroup is a triple (Xτ12,*,X) consisting of bitopological semigroups (Xτ12,*) and anS-map X:(Xτ2,*)→(Xτ1,*) from anS-space (Xτ2,*) into anS-space (Xτ1,*). In our work, for anyS-space, (Oρ,∘) can be regarded as an c-bitopological semigroup (Oρρ,∘,id) where id is the identityS-map on (Oρ,∘). That is, (Oρ,∘):=(Oρρ,∘,id).Anc-map from (Xτ12,*,X) into (Oρ,∘) is a pair f12=(f1,f2):(Xτ12,*,X)→(Oρ,∘) of an S1-map f1:(Xτ1,*)→(Oρ,∘) and S2-map f2:(Xτ2,*)→(Oρ,∘) such that f1∘X=f2.Definition 1 (see [6]).
Letf:(Xτ,*)→(Oρ,∘) and h:(Xτ′′,*′)→(Xτ,*) be twoS-maps. AnS-map f is said to have the Cℵ-fibration property by anS-map h provided for every (Yω,⋆)∈ℵ and, given twoS-maps g:(Yω,⋆)→(Xτ′′,*′) and G:(Yω,⋆)→P(Oρ,∘) with G0=f∘(h∘g), there exists anS-homotopy H:(Yω,⋆)→P(Xτ,*) such that H0=h∘g and f∘Ht=Gt for all t∈I.Definition 2 (see [6]).
Anc-map f12:(Xτ12,*,X)→(Oρ,∘) is called an Cℵ-fibration if an S1-map f1:(Xτ1,*)→(Oρ,∘) has the Cℵ-fibration property by anS-map X:(Xτ2,*)→(Xτ1,*). That is, for every (Yω,⋆)∈ℵ and given twoS-maps g:(Yω,⋆)→(Xτ2,*) and G:(Yω,⋆)→P(Oρ,∘) with G0=f2∘g, there exists anS-homotopy H:(Yω,⋆)→P(Xτ1,*) such that H0=X∘g and f1∘Ht=Gt for all t∈I.Let(Xτ12,*,B) be an c-bitopological semigroup and let (B|τ12,*) be an b-subspace of (Xτ12,*). The c-bitopological semigroup (B|τ12,*,B) is called anc-subspace of (Xτ12,*,X) provided B(b)=X(b) for all b∈B.Theorem 3 (see [6]).
Letf12:(Xτ12,*,X)→(Oρ,∘) be an c-map and (B|ρ,∘) be anS-subspace of (Oρ,∘) such that f1-1(B)=f2-1(B). Then the triple (B-|τ12,*,X|B-) is an c-subspace of (Xτ12,*,X) and a pair f12|B-=(f1|B-,f2|B-) is an c-map from an c-bitopological semigroup (B-|τ12,*,X|B-) into (B|ρ,∘), where B-=f1-1(B).Corollary 4 (see [6]).
Letf12:(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration and let (B|ρ,∘) be anS-subspace of (Oρ,∘) such that f1-1(B)=f2-1(B). Then the restriction c-map
(1)f12|B-:(B-|τ12,*,X|B-)⟶(B|ρ,∘)
is an Cℵ-fibration, where B-=f1-1(B).


## 3. The Pullback c-Maps

In this section, we show that the pullbacks ofS-maps which have the Cℵ-fibration property will also have this property and the pullbacks of Cℵ-fibrations are Cℵ-fibrations under given conditions.Letf12:(Xτ12,*,X)→(Oρ,∘) be an c-map and let h:(Qυ,⊙)→(Oρ,∘) be anS-map. Let
(2)Xhi={(x,b)∈X×Q∣fi(x)=h(b)},
where i=1,2.Lemma 5.
Letf12:(Xτ12,*,X)→(Oρ,∘) be an c-map and let h:(Qυ,⊙)→(Oρ,⊙) be anS-map. Then the pair (Xhi|τ12×υ,*×⊙) is an b-subspace of the bitopological semigroup ((X×Q)τ12×υ,*×⊙), where i=1,2.Proof.
It is clear thatXh1|τ12×υ and Xh2|τ12×υ are subspaces of a bitopological space (X×O)τ12×υ. Since h is anS-map and f1 is an S1-map, then, for all (x,b),(x′,b′)∈Xh1,
(3)h(b⊙b′)=h(b)∘h(b′)=f1(x)∘f1(x′)=f1(x*x′).
This implies
(4)(x,b)(*×⊙)(x′,b′)=(x*x′,b⊙b′)∈Xh1
for all (x,b),(x′,b′)∈Xh1. That is, (Xh1|τ12×υ,*×⊙) is an b-subspace of the bitopological semigroup ((X×O)τ12×υ,*×⊙). Similarly, (Xh2|τ12×υ,*×⊙) is an b-subspace of the bitopological semigroup ((X×O)τ12×υ,*×⊙).Henceforth, in this paper, byJ1 and J2, we mean the usual first and the second projectionS-maps (or maps), respectively.Theorem 6.
Letf12:(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration and let h:(Qυ,⊙)→(Oρ,∘) be anS-map. Then theS-map fh1:(Xh1|τ1×υ,*×⊙)→(Qυ,⊙) has the Cℵ-fibration property by anS-map Xh=X×id|Xh2 such that fh1(x,b)=b for all (x,b)∈Xh1.Proof.
Sincef12 is an c-map then, for all (x,b)∈Xh2,
(5)f1(X(x))=f2(x)=h(b).
That is, (X(x),b)∈Xh1 for all (x,b)∈Xh2. Hence, by the last lemma, Xh is a well-definedS-map taking (Xh2|τ2×υ,*×⊙) into (Xh1|τ1×υ,*×⊙).
Now let(Yω,⋆)∈ℵ and let g:(Yω,⋆)→(Xh2|τ2×υ,*×⊙) and G:(Yω,⋆)→P(Qυ,⊙) be twoS-maps with G0=fh1∘(Xh∘g).
Take anS-map g′=J1∘g:(Yω,⋆)→(Xτ2,*) and anS-homotopy
(6)G′=h∘G:(Yω,⋆)⟶P(Oρ,∘).
We observe that
(7)G′(y)(0)=h[G(y)(0)]=h[(fh1∘Xh)(g(y))]=h{fh1[X(J1(g(y))),J2(g(y))]}=h[J2(g(y))]=f2[J1(g(y))]=f2(g′(y))
for all y∈Y. That is, G0′=f2∘g′. Since f12 is an Cℵ-fibration, then there is anS-homotopy H′:(Yω,⋆)→P(Xτ1,*) such that H0′=X∘g′ and f1∘Ht′=Gt′ for all t∈I.
Define anS-homotopy H:(Yω,⋆)→P(Xh1|τ1×υ,*×⊙) by
(8)H(y)(t)=[H′(y)(t),G(y)(t)]
for all y∈Y,t∈I. We observe that fh1∘Ht=Gt for all t∈I and
(9)H(y)(0)=[H′(y)(0),G(y)(0)]=[X(g′(y)),(fh1∘Xh)(g(y))]=[X(J1(g(y))),J2(g(y))]=Xh[J1(g(y)),J2(g(y))]=Xh(g(y))=(Xh∘g)(y)
for all y∈Y. That is, H0=Xh∘g. Hence fh1 has the Cℵ-fibration property by anS-map Xh.In the last theorem, iff1=f2 (i.e., f12 is an Sp-map), let f=f1=f2; then
(10)Xh=X×id|Xh:(Xh|τ2×υ,*×⊙)⟶(Xh|τ1×υ,*×⊙)
is a well-definedS-map taking (Xh|τ2×υ,*×⊙) into (Xh|τ1×υ,*×⊙), where Xh=Xh1=Xh2. That is, the triple (Xh|τ12×υ,*×⊙,Xh) is an c-bitopological semigroup, called apullback c-bitopological semigroup of (Xτ12,*,X) induced from f12 by h. The pair
(11)f12h=(fh,fh):(Xh,τ1×υ|Xh,τ2×υ|Xh)Xh⟶(Qυ,⊙)
which is given by fh(x,b)=b for all (x,b)∈Xh is an c-map, called apullback c-map of f12 induced by h. We observe that
(12)(fh∘Xh)(x,b)=fh(X(x),b)=b=fh(x,b)
for all (x,b)∈Xh.Theorem 7.
Letf12=(f,f):(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration and let h:(Qυ,⊙)→(Oρ,∘) be anS-map such that Xh1∩Xh2≠ϕ. Then the pullback c-map f12h of f12 induced by h is an Cℵ-fibration.Proof.
It is obvious by the last theorem and the second part in Definition2.


## 4. The c-Lifting Functions

In this section, we define the path lifting property forc-maps by giving the concept of an c-lifting property and we show its role in satisfying the Cℵ-fibration property.Recall [5] that for anS-map f:(Xτ,*)→(Oρ,∘), the map: α→f∘α for all α∈P(Xτ) is anS-map from P(Xτ,*) into P(Oρ,∘), denoted by f^. Then for every c-bitopological semigroup (Xτ12,*,X), X^ is anS-map from P(Xτ2,*) into P(Xτ1,*). That is, the triple (P(X)τ12c,p(*),X^) is an c-bitopological semigroup where τ1c and τ2c are compact-open topologies on P(X) which are induced by τ1 and τ2, respectively. The shorter notion for this c-bitopological semigroup will be P(Xτ12,*,X).For a mapf:Xτ→Oρ, by Δ(f), we mean the set
(13)Δ(f)={(x,α)∈Xτ×P(Oρ)∣α(0)=f(x)}.Proposition 8.
Letf:(Xτ,*)→(Oρ,∘) be anS-map. Then (Δ(f)|τ×ρc,*×p(∘)) is anS-subspace of anS-space ((X×P(O))τ×ρc,*×p(∘)), where ρc is a compact-open topology on P(O) which is induced by ρ.Proof.
It is clear thatΔ(f)|τ×ρc is a subspace of a space (X×P(O))τ×ρc. We observe that, for all (x,α),(x′,α′)∈Δ(f),
(14)(αp(∘)α′)(0)=α(0)∘α′(0)=f(x)∘f(x′)=f(x*x′).
That is,
(15)(x,α)*×p(∘)(x′,α′)=(x*x′,αp(∘)α′)∈Δ(f).
Hence (Δ(f)|τ×ρc,*×p(∘)) is anS-subspace of anS-space ((X×P(O))τ×ρc,*×p(∘)).In the last theorem, the shorter notion for theS-space (Δ(f)|τ×ρc,*×p(∘)) will be Δ(f)|τ×ρc.Definition 9.
Letf12:(Xτ12,*,X)→(Oρ,∘) be an c-map. AnS-map
(16)L:Δ(f2)|τ2×ρc⟶P(Xτ1,*)
from anS-space Δ(f2)|τ2×ρc into P(Xτ1,*) is called anc-lifting function for an c-map f12 provided L satisfies the following: (1)
L(x,α)(0)=X(x) for all (x,α)∈Δ(f2);(2)
f1∘L(x,α)=α for all (x,α)∈Δ(f2).
Andλf will be denoted to c-lifting function for an c-map f12, if it exists.Example 10.
Let(Xτ12,*,X) be an c-bitopological semigroup. For everyS-space (Oρ,∘), the Sp-map
(17)f12=(f,f):((X×O)τ12×ρ,*×∘,X×id)⟶(Oρ,∘)
is an c-map, where f(x,y)=y for all x∈X,y∈O. Note that
(18)[f∘(X×id)](x,y)=f(X(x),y)=y=f(x,y)
for all x∈X, y∈O. This c-map has an c-lifting function
(19)λf:Δ(f)|(τ2×ρ)×ρc⟶P((X×O)τ1×ρ,τ1×∘)
which is given by
(20)λf((x,b),α)(t)=(X(x),α(t))∀((x,b),α)∈Δ(f),t∈I.
Note that
(21)λf((x,b),α)(0)=(X(x),α(0))=(X(x),f(x,b))=(X(x),b)=(X×id)(x,b),[f∘λf((x,b),α)](t)=f(X(x),α(t))=α(t)
for all ((x,b),α)∈Δ(f), t∈I.The following theorem clarifies the existence property forc-lifting function in Cℵ-fibration theory. That is, it clarifies that the existence of c-lifting function for any Cℵ-fibration is necessary and sufficient condition.Theorem 11.
Anc-map f12:(Xτ12,*,X)→(Oρ,∘) is an Cℵ-fibration if and only if there exists an c-lifting function for f12.Proof.
Suppose thatf12 is an Cℵ-fibration. Take Δ(f2)|τ2×ρc∈ℵ. Define twoS-maps
(22)g:Δ(f2)|τ2×ρc⟶(Xτ2,*),G:Δ(f2)|τ2×ρc⟶P(Oρ,∘)
by g(x,α)=x and G(x,α)=α for all (x,α)∈Δ(f2), respectively. We observe that
(23)G(x,α)(0)=α(0)=f2(x)=(f2∘g)(x,α).
Since f12 is an Cℵ-fibration, then there exists anS-homotopy H:Δ(f2)|τ2×ρc→P(Xτ1,*) such that H0=X∘g and f1∘Ht=Gt for all t∈I. Define anS-map
(24)λf:Δ(f2)|τ2×ρc⟶P(Xτ1,*)
by
(25)λf(x,α)(t)=H(x,α)(t)∀(x,α)∈Δ(f2).
We observe that, for all (x,α)∈Δ(f2),
(26)λf(x,α)(0)=H(x,α)(0)=(X∘g)(x,α)=X(x)f1∘λf(x,α)=f1∘H(x,α)=G(x,α)=α.
That is, λf is an c-lifting function for f12.
Conversely, suppose that there exists anc-lifting function λf for f12. Let (Yω,⋆)∈ℵ and let g:(Yω,⋆)→(Xτ2,*) and G:(Yω,⋆)→P(Oρ,∘) be two givenS-maps with G0=f2∘g. Define anS-homotopy H:(Yω,⋆)→P(Xτ1,*) by
(27)H(y)(t)=λf[g(y),G(y)](t)∀y∈Y,t∈I.
We observe that
(28)H(y)(0)=λf[g(y),G(y)](0)=(X∘g)(y),f1[H(y)(t)]=f1[λf[g(y),G(y)](t)]=G(y)(t)
for all y∈Y, t∈I. That is, H0=X∘g and f1∘Ht=Gt for all t∈I. Hence f12 is an Cℵ-fibration.Theorem 12.
Letf12:(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration. Then the c-map
(29)f^12=(f^1,f^2):P(Xτ12,*,X)⟶P(Oρ,∘)
is an Cℵ-fibration.Proof.
Sincef12 is an Cℵ-fibration, then there exists c-lifting function
(30)λf:Δ(f2)|τ2×ρc⟶P(Xτ1,*)
for f12 such that
(31)λf(x,α)(0)=X(x),f1∘λf(x,α)=α
for all (x,α)∈Δ(f2). Let (Yω,⋆)∈ℵ and let g:(Yω,⋆)→P(Xτ2,*) and G:(Yω,⋆)→P[P(O)ρc,p(∘)] be two givenS-maps with
(32)[G(y)(s)](0)=(f^2∘g)(y)(s)
for all y∈Y, s∈I, where ρc is a compact-open topology on P(O) which is induced by ρ. Define anS-homotopy H:(Yω,⋆)→P[P(X)τ1c,p(*)] by
(33)[H(y)(s)](t)=λf[g(y)(s),G(y)(s)](t)∀y∈Y,s,t∈I.
We observe that
(34)[H(y)(s)](0)=λf[g(y)(s),G(y)(s)](0)=X[g(y)(s)]=(X^∘g)(y)(s),(f^1∘Hr)(y)(t)=(f1∘λf[g(y)(s),G(y)(s)])(t)=[G(y)(s)](t),
for all y∈Y, s,t∈I. That is, H0=X^∘g and f^1∘Hs=Gs for all s∈I. Hence f^12 is an Cℵ-fibration.Anc-lifting function λf is calledregular if for every x∈Xτ2, λf(x,f2∘x~)=X(x)~, where x~ is the constant path in Xτ2 (i.e., x~(t)=x), similar for X(x)~. An Cℵ-fibration f12 is calledregular if it has regular c-lifting function.Example 13.
In Example10, the c-lifting function λf which is given by
(35)λf((x,b),α)(t)=(X(x),α(t))∀((x,b),α)∈Δ(f),t∈I
is regular. Note that, for every (x,b)∈(X×O)τ2,
(36)λf((x,b),f2∘(x,b)~)(t)=(X(x),(f2∘(x,b)~)(t))=(X(x),f2(x,b))=(X(x),b)=(X×id)(x,b)=(X×id)(x,b)~(t)
for all t∈I.The following theorem is an analogue of results of Fadell in Hurewicz fibration theory [7].Theorem 14.
Letf12:(Xτ12,*,X)→(Oρ,∘) be a regular Cℵ-fibration and let
(37)Mi:P(Xτi,*)⟶Δ(fi)|τi×ρc
be anS-map defined by Mi(α)=(α(0),fi∘α) for all α∈P(Xτi) where i=1,2. Then (1)
M1∘λf=X×id|Δ(f2);(2)
λf∘M2≃sX^ preserving projection. That is, there is anS-homotopy
(38)H:P(Xτ2,*)⟶P[P(X)τ1c,p(*)]
between twoS-maps λf∘M2 and X^ such that f1[(H(α)(s))(t)]=f2(α(t)) for all t,s∈I,α∈P(Xτ2).Proof.
For the first part, we observe that, for every(x,α)∈Δ(f2),
(39)(M1∘λf)(x,α)=M1[λf(x,α)]=(λf(x,α)(0),f1∘λf(x,α))=(X(x),α)=(X×id)(x,α).
That is, M1∘λf=X×id|Δ(f2).
For the second part, forα∈P(Xτ2) and s∈I, define a path β1-s∈P(Oρ) by
(40)β1-s(t)={f2(α(s+t)),for0≤t≤1-s,f2(α(1)),for1-s≤t≤1.
By the regularity of λf, we can define anS-homotopy H:P(Xτ2,*)→P[P(X)τ1c,p(*)] by
(41)[H(α)(s)](t)={X[α(t)],for0≤t≤s,λf(α(s),β1-s)(t-s),fors≤t≤1,
for all s∈I, α∈P(Xτ2). Then
(42)[H(α)(0)](t)=λf(α(0),β1)(t)=λf(α(0),β)(t)=λf(α(0),f2∘α)(t)=(λf∘M2)(α)(t),[H(α)(1)](t)=X[α(t)]=X^(α)(t)
for all α∈P(Xτ2), t∈I. That is, λf∘M2≃sX^. Also we get that
(43)[f1∘(H(α)(s))](t)={f1(X[α(t)]),for0≤t≤s,f1[λf(α(s),β1-s)(t-s)],fors≤t≤1;={f2(α(t)),for0≤t≤s,β1-s(t-s),fors≤t≤1;={f2(α(t)),for0≤t≤s,f2(α(s+t-s)),fors≤t≤1;={f2(α(t)),for0≤t≤s,f2(α(t)),fors≤t≤1;=f2(α(t)),
for all s,t∈I, α∈P(Xτ2). Hence λf∘M2≃sX^ preserving projection.


## 5. Approximate Fibrations

Coram and Duvall [2] introduced approximate fibrations as a generalization of cell-like maps [3] and showed that the uniform limit of a sequence of Hurewicz fibrations is an approximate fibration. A map f:Xτ→Oρ of compact metrizable spaces Xτ and Oρ is called anapproximate fibration if, for every space Yω and for given ϵ>0, there exists δ>0 such that whenever g:Yω→Xτ and H:Yω×I→Oρ are maps with d[H(y,0),(f∘g)(y)]<δ, then there is homotopy G:Yω×I→Xτ such that G0=g and
(44)d[H(y,t),(f∘G)(y,t)]<ϵ∀y∈Y,t∈I.
One notable exception is that the pullback of approximate fibration need not be an approximate fibration.The following theorem shows the role of theCNπ-fibration property in inducing an approximate fibration property.For anS-map f:(Xτ,π)→(Oρ,π) with metrizable spaces Xτ and Oρ, by dτ and dρ, we mean the metric functions on X and O, respectively; by X×O we mean the product metrizable space of Xτ and Oρ with a metric function
(45)d((x,b),(x′,b′))=max⁡{dτ(x,x′),dρ(b,b′)};
by Gf we mean the graph of f (i.e., Gf={(x,f(x)):x∈X}) which is anS-subspace of ((X×O)τ×ρ,π); for a positive integer n>0, by Gn(f), we mean the (1/n)-neighborhood of Gf in a metrizable space X×O which is alsoS-subspace of ((X×O)τ×ρ,π).Theorem 15.
Letf:Xτ→Oρ be a map with compact metrizable spaces Xτ and Oρ. Then f is an approximate fibration if and only if, for every positive integer n>0, there exists a positive integer m≥n such that theS-map fn:(Gn(f),π)→(Oρ,π) has the CNπ-fibration property by the inclusionS-map Imn:(Gm(f),π)→(Gn(f),π), where fn(x,b)=b for all (x,b)∈Gn(f).Proof.
Letn be any positive integer. For ϵ=1/n>0, let δ be given in the definition of approximate fibration. Since δ/2>0 and f is a continuous function, then let δ′ be chosen such that if x,x′∈X and dτ(x,x′)<δ′, then dρ(f(x),f(x′))<ϵ/2. Choose a positive integer m⩾n, such that 1/m≤δ′,δ/2.
Now let(Yω,π)∈Nπ and let g:(Yω,π)→(Gm(f),π) and G:(Yω,π)→P(Oρ,π) be two givenS-maps with G0=fn∘(Imn∘g). Define a map g′:Yω→Xτ by g′(y)=J1[g(y)] and a homotopy G′:Yω×I→Oρ by G′(y,t)=G(y)(t) for all y∈Y and t∈I. We get that g(y)=(g′(y),G′(y,0)) for all y∈Y. Since g(y)∈Gm(f), then there exists x∈X such that
(46)d[(x,f(x)),g(y)]<1m.
Then
(47)dτ(x,g′(y))<1m⩽δ′,dρ(f(x),G′(y,0))<1m⩽δ2,dρ(f(x),f(g′(y)))<1m⩽δ2
for all y∈Y. This implies
(48)dρ(f(g′(y)),G′(y,0))⩽dρ(f(g′(y)),f(x))+dρ(f(x),G′(y,0))<δ
for all y∈Y. Hence, since f is an approximate fibration, there exists a homotopy H′:Yω×I→Xτ such that H0′=g′ and
(49)dτ(G′(y,t),(f∘Ht′)(y))<ϵ
for all y∈Y, t∈I. Define anS-homotopy H:(Yω,π)→P(Gn(f),π) by
(50)H(y)(t)=(H′(y,t),G(y)(t))∀y∈Y,t∈I.
Then we get that
(51)H(y)(0)=(H′(y,0),G(y)(0))=(g′(y),G(y)(0))=g(y)=(Imn∘g)(y)
for all y∈Y and fn∘Ht=Gt for all t∈I. Hence fn has the CNπ-fibration property by Imn.
Conversely, letϵ>0 be given. Since f is a continuous function, then let δ′ be chosen such that if x,x′∈X and dτ(x,x′)<δ′, then dρ(f(x),f(x′))<ϵ/2. Choose a positive integer n>0 such that 1/n≤δ′,ϵ/2. By hypothesis, there exists a positive integer m≥n such that fn has the CNπ-fibration property by Imn.
Takeδ=1/m. Let Yω be any space and let g:Yω→Xτ and G:Yω×I→Oρ be two given maps with
(52)dρ[G(y,0),(f∘g)(y)]<δ
for all y∈Y. Define anS-map g′:(Yω,π)→(Gm(f),π) by g′(y)=(g(y),G(y,0)) and anS-homotopy G′:(Yω,π)→P(Oρ,ı) by G′(y)(t)=G(y,t) for all y∈Y and t∈I. Since G0′=fn∘(Imn∘g′), then there exists anS-homotopy F:(Yω,π)→P(Gn(f),π) such that F0=Imn∘g′ and fn∘Ft=Gt′ for all t∈I. By the last part, we can define a homotopy H:Yω×I→Xτ by
(53)H(y,t)=J1[F(y)(t)]∀y∈Y,t∈I.
We get that F(y)(t)=(H(y,t),G(y,t)). Since F(y)(t)∈Gn(f), then there exists x∈X such that
(54)d[(x,f(x)),F(y)(t)]<1n.
Then
(55)dτ(x,H(y,t))<1n⩽δ′,dρ(f(x),G(y,t))<1n⩽ϵ2,dρ[f(x),f(H(y,t))]<1n⩽ϵ2.
This implies
(56)dρ[G(y,t),f(H(y,t))]⩽dρ[f(H(y,t)),f(x)]+dρ(f(x),G(y,t))<ϵ
for all y∈Y,t∈I. Hence f is an approximate fibration.



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*Source: 675761-2014-08-17.xml*