#### Abstract

We extend the path lifting property in homotopy theory for topological spaces to bitopological semigroups and we show and prove its role in the -fibration property. We give and prove the relationship between the -fibration property and an approximate fibration property. Furthermore, we study the pullback maps for -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 -fibrations as extension of Hurewicz fibrations. In [6], we introduced the concepts of bitopological semigroups, -bitopological semigroups, and -fibrations as extension of -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 -maps which have the -fibration property that will also have this property and the pullbacks of -fibrations are -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 -fibration property and an approximate fibration property.

#### 2. Preliminaries

Throughout this paper, by all we mean all topological spaces which will be assumed Hausdorff spaces. By all we mean all bitopological spaces . For two bitopological spaces and , a* p-map * is a function from into that is continuous function (i.e., a map) from a space into a space and from into [4].

Recall [5] that a* topological semigroup* or an* S-space* is a pair consisting of a topological space and a map from the product space into such that for all . An* S*-space is called an* S-subspace* of if is a subspace of and the map takes the product into and for all . We denote the class of all* S*-spaces by . For every space , by , we mean the space of all paths from the unit closed interval into with the compact-open topology. Recall [5] that, for every* S*-space , is an* S*-space where is a map defined by for all , . The shorter notion for this* S*-space will be . For every space , the* natural S-space* is an* S*-space , where is a continuous associative multiplication on given by and for all . We denote the class of all natural* S*-spaces by , where .

Recall [5] that the function is called an* S-map* if is a map of a space into and for all . The function of a natural* S*-space into is an* S*-map if and only if it is continuous. The* S*-maps are called* S-homotopic* and write provided there is an* S*-map called an* S-homotopy* such that and for all .

A* bitopological semigroup* is a pair consisting of a bitopological space and the associative multiplication on such that is an -map from the product bitopological space into . For , by we mean the bitopological subspace of . If the -map takes the product into then the pair will be a bitopological semigroup and will be called an* b-subspace* of .

The function is called an -*map* from into provided is an* S*-map from a function* S*-space into an* S*-space , where . We say that is an* Sp-map* if it is an -map and -map.

An* c-bitopological semigroup* is a triple consisting of bitopological semigroups and an* S*-map from an* S*-space into an* S*-space . In our work, for any* S*-space, can be regarded as an -bitopological semigroup where is the identity* S*-map on . That is, .

An* c-map* from into is a pair of an -map and -map such that .

*Definition 1 (see [6]). *Let and be two* S*-maps. An* S*-map is said to have the -fibration property by an* S*-map provided for every and, given two* S*-maps and with , there exists an* S*-homotopy such that and for all .

*Definition 2 (see [6]). *An -map is called an -fibration if an -map has the -fibration property by an* S*-map . That is, for every and given two* S*-maps and with , there exists an* S*-homotopy such that and for all .

Let be an c-bitopological semigroup and let be an -subspace of . The -bitopological semigroup is called an* c-subspace* of provided for all .

Theorem 3 (see [6]). *Let be an c-map and be an S-subspace of such that . Then the triple is an c-subspace of and a pair is an c-map from an c-bitopological semigroup into , where .*

Corollary 4 (see [6]). *Let be an -fibration and let be an S-subspace of such that . Then the restriction c-map
*

*is an -fibration, where .*

#### 3. The Pullback c-Maps

In this section, we show that the pullbacks of* S*-maps which have the -fibration property will also have this property and the pullbacks of -fibrations are -fibrations under given conditions.

Let be an -map and let be an* S*-map. Let
where .

Lemma 5. *Let be an c-map and let be an S-map. Then the pair is an b-subspace of the bitopological semigroup , where .*

*Proof. *It is clear that and are subspaces of a bitopological space . Since is an* S*-map and is an -map, then, for all ,
This implies
for all . That is, is an -subspace of the bitopological semigroup . Similarly, is an -subspace of the bitopological semigroup .

Henceforth, in this paper, by and , we mean the usual first and the second projection* S*-maps (or maps), respectively.

Theorem 6. *Let be an -fibration and let be an S-map. Then the S-map has the -fibration property by an S-map such that for all .*

*Proof. *Since is an -map then, for all ,
That is, for all . Hence, by the last lemma, is a well-defined* S*-map taking into .

Now let and let and be two* S*-maps with .

Take an* S*-map and an* S*-homotopy
We observe that
for all . That is, . Since is an -fibration, then there is an* S*-homotopy such that and for all .

Define an* S*-homotopy by
for all . We observe that for all and
for all . That is, . Hence has the -fibration property by an* S*-map .

In the last theorem, if (i.e., is an -map), let ; then
is a well-defined* S*-map taking into , where . That is, the triple is an -bitopological semigroup, called a* pullback c-bitopological semigroup* of induced from by . The pair
which is given by for all is an -map, called a* pullback c-map* of induced by . We observe that
for all .

Theorem 7. *Let be an -fibration and let be an S-map such that . Then the pullback -map of induced by is an -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 -maps by giving the concept of an -lifting property and we show its role in satisfying the -fibration property.

Recall [5] that for an* S*-map , the map: for all is an* S*-map from into , denoted by . Then for every -bitopological semigroup , is an* S*-map from into . That is, the triple is an -bitopological semigroup where and are compact-open topologies on which are induced by and , respectively. The shorter notion for this c-bitopological semigroup will be .

For a map , by , we mean the set

Proposition 8. *Let be an S-map. Then is an S-subspace of an S-space , where is a compact-open topology on which is induced by .*

*Proof. *It is clear that is a subspace of a space . We observe that, for all ,
That is,
Hence is an* S*-subspace of an* S*-space .

In the last theorem, the shorter notion for the* S*-space will be .

*Definition 9. *Let be an -map. An* S*-map
from an* S*-space into is called an* c-lifting function* for an -map provided satisfies the following: (1) for all ;(2) for all .

And will be denoted to -lifting function for an -map , if it exists.

*Example 10. *Let be an -bitopological semigroup. For every* S*-space , the -map
is an -map, where for all . Note that
for all , . This -map has an -lifting function
which is given by
Note that
for all , .

The following theorem clarifies the existence property for -lifting function in -fibration theory. That is, it clarifies that the existence of -lifting function for any -fibration is necessary and sufficient condition.

Theorem 11. *An -map is an -fibration if and only if there exists an -lifting function for .*

*Proof. *Suppose that is an -fibration. Take . Define two* S*-maps
by and for all , respectively. We observe that
Since is an -fibration, then there exists an* S*-homotopy such that and for all . Define an* S*-map
by
We observe that, for all ,
That is, is an -lifting function for .

Conversely, suppose that there exists an -lifting function for . Let and let and be two given* S*-maps with . Define an* S*-homotopy by
We observe that
for all , . That is, and for all . Hence is an -fibration.

Theorem 12. *Let be an -fibration. Then the c-map
**
is an -fibration.*

*Proof. *Since is an -fibration, then there exists -lifting function
for such that
for all . Let and let and be two given* S*-maps with
for all , , where is a compact-open topology on which is induced by . Define an* S*-homotopy by
We observe that
for all , . That is, and for all . Hence is an -fibration.

An -lifting function is called* regular* if for every , , where is the constant path in (i.e., ), similar for . An -fibration is called* regular* if it has regular c-lifting function.

*Example 13. *In Example 10, the -lifting function which is given by
is regular. Note that, for every ,
for all .

The following theorem is an analogue of results of Fadell in Hurewicz fibration theory [7].

Theorem 14. *Let be a regular -fibration and let
**
be an S-map defined by for all where . Then *(1)

*;*(2)

*preserving projection. That is, there is an*

*S*-homotopy between two*S*-maps and such that for all .*Proof. *For the first part, we observe that, for every ,
That is, .

For the second part, for and , define a path by
By the regularity of , we can define an* S*-homotopy by
for all , . Then
for all , . That is, . Also we get that
for all , . Hence 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 of compact metrizable spaces and is called an* approximate fibration* if, for every space and for given , there exists such that whenever and are maps with , then there is homotopy such that and
One notable exception is that the pullback of approximate fibration need not be an approximate fibration.

The following theorem shows the role of the -fibration property in inducing an approximate fibration property.

For an* S*-map with metrizable spaces and , by and , we mean the metric functions on and , respectively; by we mean the product metrizable space of and with a metric function
by we mean the graph of (i.e., ) which is an* S*-subspace of ; for a positive integer , by , we mean the -neighborhood of in a metrizable space which is also* S*-subspace of .

Theorem 15. *Let be a map with compact metrizable spaces and . Then is an approximate fibration if and only if, for every positive integer , there exists a positive integer such that the S-map has the -fibration property by the inclusion S-map , where for all .*

*Proof. *Let be any positive integer. For , let be given in the definition of approximate fibration. Since and is a continuous function, then let be chosen such that if and , then . Choose a positive integer , such that .

Now let and let and be two given* S*-maps with . Define a map by and a homotopy by for all and . We get that for all . Since , then there exists such that
Then
for all . This implies
for all . Hence, since is an approximate fibration, there exists a homotopy such that and
for all , . Define an* S*-homotopy by
Then we get that
for all and for all . Hence has the -fibration property by .

Conversely, let be given. Since is a continuous function, then let be chosen such that if and , then . Choose a positive integer such that . By hypothesis, there exists a positive integer such that has the -fibration property by .

Take . Let be any space and let and be two given maps with
for all . Define an* S*-map by and an* S*-homotopy by for all and . Since , then there exists an* S*-homotopy such that and for all . By the last part, we can define a homotopy by
We get that . Since , then there exists such that
Then
This implies
for all . Hence 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.