We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the Cℵ-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.

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, Section 2 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 all Xτ 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, a p-map h: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 a topological semigroup or an S-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. An S-space (A,*′) is called an S-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 all S-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 every S-space (Xτ,*), (P(Xτ),p(*)) is an S-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 this S-space will be P(Xτ,*). For every space Xτ, the natural S-space is an S-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 natural S-spaces (Xτ,π) by Nπ, where π=π1,π2.

Recall [5] that the function f:(Xτ,*)→(Oρ,∘) is called an S-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 natural S-space (Xτ,π) into (Oρ,π) is an S-map if and only if it is continuous. The S-maps f,g:(Xτ,*)→(Oρ,∘) are called S-homotopic and write f≃sg provided there is an S-map H:(Xτ,*)→P(Oρ,∘) called an S-homotopy such that H(x)(0)=f(x) and H(x)(1)=g(x) for all x∈X.

A bitopological 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 an b-subspace of (Xτ12,*).

The function h:(Xτ12,*)→(Yρ12,∘) is called an Si-map from (Xτ12,*) into (Yρ12,∘) provided h is an S-map from a function S-space (Xτi,*) into an S-space (Yρi,∘), where i=1,2. We say that h is an Sp-map if it is an S1-map and S2-map.

An c-bitopological semigroup is a triple (Xτ12,*,X) consisting of bitopological semigroups (Xτ12,*) and an S-map X:(Xτ2,*)→(Xτ1,*) from an S-space (Xτ2,*) into an S-space (Xτ1,*). In our work, for any S-space, (Oρ,∘) can be regarded as an c-bitopological semigroup (Oρρ,∘,id) where id is the identity S-map on (Oρ,∘). That is, (Oρ,∘):=(Oρρ,∘,id).

An c-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 [<xref ref-type="bibr" rid="B4">6</xref>]).

Let f:(Xτ,*)→(Oρ,∘) and h:(Xτ′′,*′)→(Xτ,*) be two S-maps. An S-map f is said to have the Cℵ-fibration property by an S-map h provided for every (Yω,⋆)∈ℵ and, given two S-maps g:(Yω,⋆)→(Xτ′′,*′) and G:(Yω,⋆)→P(Oρ,∘) with G0=f∘(h∘g), there exists an S-homotopy H:(Yω,⋆)→P(Xτ,*) such that H0=h∘g and f∘Ht=Gt for all t∈I.

Definition 2 (see [<xref ref-type="bibr" rid="B4">6</xref>]).

An c-map f12:(Xτ12,*,X)→(Oρ,∘) is called an Cℵ-fibration if an S1-map f1:(Xτ1,*)→(Oρ,∘) has the Cℵ-fibration property by an S-map X:(Xτ2,*)→(Xτ1,*). That is, for every (Yω,⋆)∈ℵ and given two S-maps g:(Yω,⋆)→(Xτ2,*) and G:(Yω,⋆)→P(Oρ,∘) with G0=f2∘g, there exists an S-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 an c-subspace of (Xτ12,*,X) provided B(b)=X(b) for all b∈B.

Theorem 3 (see [<xref ref-type="bibr" rid="B4">6</xref>]).

Let f12:(Xτ12,*,X)→(Oρ,∘) be an c-map and (B|ρ,∘) be an S-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 [<xref ref-type="bibr" rid="B4">6</xref>]).

Let f12:(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration and let (B|ρ,∘) be an S-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 of S-maps which have the Cℵ-fibration property will also have this property and the pullbacks of Cℵ-fibrations are Cℵ-fibrations under given conditions.

Let f12:(Xτ12,*,X)→(Oρ,∘) be an c-map and let h:(Qυ,⊙)→(Oρ,∘) be an S-map. Let
(2)Xhi={(x,b)∈X×Q∣fi(x)=h(b)},
where i=1,2.

Lemma 5.

Let f12:(Xτ12,*,X)→(Oρ,∘) be an c-map and let h:(Qυ,⊙)→(Oρ,⊙) be an S-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 that Xh1|τ12×υ and Xh2|τ12×υ are subspaces of a bitopological space (X×O)τ12×υ. Since h is an S-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, by J1 and J2, we mean the usual first and the second projection S-maps (or maps), respectively.

Theorem 6.

Let f12:(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration and let h:(Qυ,⊙)→(Oρ,∘) be an S-map. Then the S-map fh1:(Xh1|τ1×υ,*×⊙)→(Qυ,⊙) has the Cℵ-fibration property by an S-map Xh=X×id|Xh2 such that fh1(x,b)=b for all (x,b)∈Xh1.

Proof.

Since f12 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-defined S-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 two S-maps with G0=fh1∘(Xh∘g).

Take an S-map g′=J1∘g:(Yω,⋆)→(Xτ2,*) and an S-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 an S-homotopy H′:(Yω,⋆)→P(Xτ1,*) such that H0′=X∘g′ and f1∘Ht′=Gt′ for all t∈I.

Define an S-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 an S-map Xh.

In the last theorem, if f1=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-defined S-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 a pullback 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 a pullback 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.

Let f12=(f,f):(Xτ12,*,X)→(Oρ,∘) be an Cℵ-fibration and let h:(Qυ,⊙)→(Oρ,∘) be an S-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 Definition 2.

4. The c-Lifting Functions

In this section, we define the path lifting property for c-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 an S-map f:(Xτ,*)→(Oρ,∘), the map: α→f∘α for all α∈P(Xτ) is an S-map from P(Xτ,*) into P(Oρ,∘), denoted by f^. Then for every c-bitopological semigroup (Xτ12,*,X), X^ is an S-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 map f:Xτ→Oρ, by Δ(f), we mean the set
(13)Δ(f)={(x,α)∈Xτ×P(Oρ)∣α(0)=f(x)}.

Proposition 8.

Let f:(Xτ,*)→(Oρ,∘) be an S-map. Then (Δ(f)|τ×ρc,*×p(∘)) is an S-subspace of an S-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 that Δ(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 an S-subspace of an S-space ((X×P(O))τ×ρc,*×p(∘)).

In the last theorem, the shorter notion for the S-space (Δ(f)|τ×ρc,*×p(∘)) will be Δ(f)|τ×ρc.

Definition 9.

Let f12:(Xτ12,*,X)→(Oρ,∘) be an c-map. An S-map
(16)L:Δ(f2)|τ2×ρc⟶P(Xτ1,*)
from an S-space Δ(f2)|τ2×ρc into P(Xτ1,*) is called an c-lifting function for an c-map f12 provided L satisfies the following:

L(x,α)(0)=X(x) for all (x,α)∈Δ(f2);

f1∘L(x,α)=α for all (x,α)∈Δ(f2).

And λf 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 every S-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 for c-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.

An c-map f12:(Xτ12,*,X)→(Oρ,∘) is an Cℵ-fibration if and only if there exists an c-lifting function for f12.

Proof.

Suppose that f12 is an Cℵ-fibration. Take Δ(f2)|τ2×ρc∈ℵ. Define two S-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 an S-homotopy H:Δ(f2)|τ2×ρc→P(Xτ1,*) such that H0=X∘g and f1∘Ht=Gt for all t∈I. Define an S-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 an c-lifting function λf for f12. Let (Yω,⋆)∈ℵ and let g:(Yω,⋆)→(Xτ2,*) and G:(Yω,⋆)→P(Oρ,∘) be two given S-maps with G0=f2∘g. Define an S-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.

Let f12:(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.

Since f12 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 given S-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 an S-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.

An c-lifting function λf is called regular 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 called regular if it has regular c-lifting function.

Example 13.

In Example 10, 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.

Let f12:(Xτ12,*,X)→(Oρ,∘) be a regular Cℵ-fibration and let
(37)Mi:P(Xτi,*)⟶Δ(fi)|τi×ρc
be an S-map defined by Mi(α)=(α(0),fi∘α) for all α∈P(Xτi) where i=1,2. Then

M1∘λf=X×id|Δ(f2);

λf∘M2≃sX^ preserving projection. That is, there is an S-homotopy
(38)H:P(Xτ2,*)⟶P[P(X)τ1c,p(*)]
between two S-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, for α∈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 an S-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 an approximate 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 the CNπ-fibration property in inducing an approximate fibration property.

For an S-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 an S-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 also S-subspace of ((X×O)τ×ρ,π).

Theorem 15.

Let f: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 the S-map fn:(Gn(f),π)→(Oρ,π) has the CNπ-fibration property by the inclusion S-map Imn:(Gm(f),π)→(Gn(f),π), where fn(x,b)=b for all (x,b)∈Gn(f).

Proof.

Let n 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 given S-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 an S-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, let ϵ>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.

Take δ=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 an S-map g′:(Yω,π)→(Gm(f),π) by g′(y)=(g(y),G(y,0)) and an S-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 an S-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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors also gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. 5527068.

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