1. Introduction
An algebraically chained topological space is essentially a connected topological space endowed with algebraic ordering relations. The topological notion of connectedness was first investigated by Bolzano, which was based on continua (i.e., considering algebraic ordering) [
1]. The equivalence between the notions of continuum of real space as a chain and the connectedness in Euclidean space was proposed by Bolzano. Note that the continuum of the set of real numbers
is enriched with an algebraic linear-ordering relation forming a chain such that
becomes an ordered field under the algebraic linear order
. Recall that the concept of continuum and compact chain can be formulated in a metric space
, where
is connected and compact [
2,
3]. It was noted by Lefschetz that any compact metric space can admit topological chains as well as chain complexes, where the concept of
retains the topological notion of local connectedness [
4]. Moreover, a continuum
is chainable by
if it covers the space
[
2]. Later Cantor pointed out that the definition of topological connectedness given by Bolzano could lead to a set of continuums
such that
, where
is an index set [
1]. As a result, Cantor further refined the concept of connectedness by separating the notion of (chain like) continuum of real space from the topological notion of connectedness as follows [
5].
Definition 1. If is a Euclidean space then is connected and there is a sequence such that the following three conditions are maintained by a function within the space: (1) , (2) and, (3) .
Evidently, Cantor is considering that
is a metric space (not necessarily compact) and it leads to the notion of a perfect set where
is closed and dense in itself. Later Jordan incorporated the concept of algebraic chains and topologically connected chain-components as follows [
1].
Definition 2. A topological subspace is a connected component if such that . Moreover, in if such that then is a component.
Observe that an algebraically chained topological space is essentially a connected space endowed with additional order structures. For example, the Riesz topological space is a partially ordered vector space equipped with a lattice ordered structure. The Riesz topological space invites the concept of sequential convergence and connectedness, which can be presented as follows [
6].
Definition 3. Let be an (Archimedean) Riesz space with the weak order and . The sequence converges relatively to if the sequence admits .
It is straightforward to observe that if a Riesz topological space is sequentially convergent within the space then it is (sequentially) connected. It is important to note that the earlier notions of topological connectedness employed closed sets and the Riesz ordered space moves away from such a notion of connectedness. On the contrary, Riesz proposed an alternative definition of topological connectedness and order types [
7].
Definition 4. A topological space is said to be connected if it cannot be decomposed into such that and . Moreover, a space can be an ordered space.
Evidently, the concept of Riesz connectedness of a topological space considers that
and , in general. The distinction between Cantor and Riesz is that the Riesz topological connectedness does not employ the notion of metric space. Finally, Hausdorff further generalized the concept of separation of a topological space (without considering chains) as the following:
A topological space is connected if and . Thus, the Hausdorff connectedness does not resort to any additional algebraic order types. On the other hand, Lefschetz had shown that a chainable, topologically connected and metrizable space does not contain any isolated point [
4]. Moreover, the embeddings of a Nachbin preordered set
generate Alexandroff topology endowed with topological order
admitting the following property (often called a Nachbin topological property) [
8].
Definition 5. The full embedding of a preordered set forms Alexandroff topological order such that , where and denote respective neighborhood filters and the relation denotes that is a topologically finer neighborhood filter than .
Note that a preordering relation is reflexive and transitive without paying any attention to the symmetry/asymmetry property and a preordered set can be separated. Thus, there are interactions between the concepts of topological chains, ordered spaces, connectedness and the separation of topological spaces.
1.1. Motivation
The constructions of topological spaces based on multiple order types are interesting because the resulting topological subspaces can be enriched with additional algebraic properties. In 1905 Riesz proposed that any arbitrary set
can be equipped with
different algebraic orders and such a space can be denoted as
[
7]. For example, if
such that
, then
is enriched with
types where
and
. In general, two sets
are endowed with same algebraic order if there is a bijection
preserving the algebraic order relation. However, this is a stronger condition for retaining ordering relations and it can be relaxed. Earlier, it was shown that two topological hyperconvex spaces can be ordered under a special topological-ordering relation affecting the measurability in such spaces [
9]. A topological space
is Bolzano–Weierstrass space (
) if every infinite subset of
has an accumulation point. In general, the abstract (differentiable) manifold theory is often constructed on
topological spaces and the chains in such a topological space are formulated by employing a family of sets within the corresponding topological spaces [
5,
10]. Note that such topological chained spaces are not enriched with any additional algebraic ordering relations. It is shown by Corson that every
linear (weak) topological space admits countable chains [
11,
12]. In the special topological spaces, such as tree space
, the chains can be admitted and such chains as well as their anti-chains are countable [
11]. However, these are specific topological spaces, namely weak topological Banach spaces, containing chains and the results may not be applicable to the general topological spaces. However, the interesting topological property of the chained spaces is that every para-compact space with countable chains is Lindelöf and there are relations between the open covering and topologically connected chains [
5,
11]. Urysohn, showed that real-valued continuous functions can be considered as ordered chains and moreover, a set of topological objects, called separating chains, can be employed to define a set of continuous real-valued functions [
13]. Furthermore, the chained components can be formed within a topological space under an algebraic equivalence relation [
14]. These observations provide motivation to ask the following questions towards generalizations: (1) What are the topological properties of covering spaces of chart maps of a foliated
n-manifold containing algebraic chains with different directions and orientations, (2) What are the properties of various chain-paths constructed under bijections in covering spaces in view of path-homotopy theory of algebraic topology, and (3) How to formulate an algebraically ordered chained covering space without resorting to a topological inclusion-order relation of open sets? This paper addresses these questions in relative detail in the views of general topology as well as algebraic topology.
1.2. Contributions
The contributions made in this paper can be summarized as follows: Inspired by Riesz ordered multidimensional spaces , Jordan chain components and the observations by Urysohn about real-valued functions, we propose a set of covering spaces of chart maps of a foliated n-manifold where . A connected n-space chart (topological subspace) generated by is finitely chained under algebraic ordering as , where denotes an order relation in an n-space. The covering maps of the chained n-space charts are formulated to form two varieties of algebraically chained subspaces: (1) the simply directed chains, and (2) the twisted chains under surjections within the covering spaces. Accordingly, two varieties of homotopy chained-paths are formed within the covering spaces under bijections by following the principles of: (a) path-homotopy of algebraic topology, and (b) combinatorial split–join of multiple homotopy chain-paths on the discrete fiber space. The resulting homotopy chain-paths under split–join in the covering spaces are either simply directed or oriented in nature. We show that the embedding of a topological chained subspace in is unique, while maintaining the algebraic order, and any non-homogeneous chain-paths cannot be embedded in . Interestingly, two split–join homotopy chain-paths do not mutually commute. This is to emphasize that the chains in covering spaces, as proposed in this paper, do not form any nerves homeomorphic to and the algebraic relation is not an equivalence relation forming any equivalence classes within the covering spaces, maintaining generality. Moreover, we show that all algebraically chained Hausdorff subspaces within a covering space can be considered as a measure-space and it is finitely measurable. As a result, different measures give rise to two varieties of group algebraic structures. It forms a 5th order multiplicative group if the measure in Hausdorff covering space is not unique and it forms a 6th order cyclic group if the integer-valued finite measure is unique in a Hausdorff covering space. Finally, this is to emphasize that the proposed chain-homotopy does not depend on the concepts of separating chains and inclusion-order of open sets in a topological space. Furthermore, we do not follow the formulation of Nachbin preordered set embeddings to form Alexandroff topological order in the covering spaces and it does not affect measurability.
The rest of the paper is organized as follows. The preliminary concepts related to the domain are presented in
Section 2. The definitions of split–join chains and homotopy chain-paths are presented in
Section 3.
Section 4 presents the main results. Finally,
Section 5 concludes the paper.
2. Preliminaries
In this section, the concepts and definitions of topological chains, chain connected spaces and associated algebraic structures are introduced. The classical topological properties of cover of chained-subspaces, chain-homeomorphism and separations are illustrated. Note that, the index set is denoted as
and the sets
denote the sets of real numbers and integers, respectively. The homeomorphism between
and
is denoted as
. First, we present the definition of a topological chain within a space [
5].
Definition 6. Let be a sequence in a topological space . If the sequence maintains the property that then is a topological chain in .
As a consequence, if
and
then the pair
is said to be topologically chain-connected in
. Moreover, a topological space
is a connected space if
where
is a cover of
and
. We can further extend this concept towards the formation of nerves in a chained topological space as follows [
3].
Definition 7. If is a covering chain in the topological space then it is a circular chain if the nerve of is homeomorphic to .
Remark 1. If there is a non-decomposable and non-chainable continuum, which is homeomorphic to then it is called a pseudo-solenoid, and a planar continuum is always homeomorphic to a pseudo-circle in a topological space [3]. The algebraic relation induced within a topological space and the open covers of the corresponding topological space lead to the formation of two varieties of topological chains, which are defined as follows [
3,
14].
Definition 8. Let be an equivalence relation. The equivalence classes generated by within a topological space are called and a sequence is a finite . Moreover, if is a continuum with open cover endowed with a circular nerve homeomorphic to then forms a in .
This leads to the following lemma involving the covering map in a pseudo-solenoid space [
3].
Lemma 1. Ifis a finite covering map andis continuum then ifis pseudo-solenoid then it implies thatis also a pseudo-solenoid.
Proof. It is straightforward to observe that the covering map is a surjection preserving the properties given as: and , where and are identity functions in the respective topological spaces retaining local homeomorphisms. □
Note that if we consider that the base space
is not a pseudo-solenoid, then the covering map
can admit a pseudo-solenoid structure in the covering space
if and only if the covering map preserves the
property. Recall that a topological space
is called an
if any arbitrary sets
the space maintains the topological property that:
considering that
. If we consider two
of
then a covering homeomorphism between the corresponding chained-spaces can be found, which is presented in the following theorem [
15].
Theorem 1. If is a continuous surjection between two then it is a covering if such that is a chain-homeomorphism and .
It is important to note that, in this case the topological chain-subspaces are finite in nature and the proof of the theorem is given in [
15]. The algebraic chains within a topological space are often constructed from the partially ordering relation. Interestingly, in a
partially ordered topological space, the locally compact subspaces are separated by upper and lower chain-points [
16]. This is to emphasize that a linearly ordered
topological space retains the properties of partially ordered chaining properties of the corresponding topological space. A topological space
is equipped with countable-chain-condition if
the family of sets are countable such that
and
condition is preserved [
17]. The topological chains formulated under inclusion-order of open sets leads to the definition of separating chains [
13].
Definition 9. Let us consider the countable set in a topological space . A set of topological chains under inclusion-order is called a separating chain if it maintains the following three properties, which are given as:
- (1)
,
- (2)
- (3)
As a result, the topological compactness can be conceptualized by means of separating chains as presented in the following theorem [
13].
Theorem 2. A topological space is pseudo-compact if every separating chain in contains .
The indexing schemes of separating chains and the proof of the theorem are detailed in [
13]. It is important to note that a set of separating chains can help in determining the separation of a topological space. On the other hand, if we consider a partially ordered set
generating algebraic
, then a topological space
containing
satisfies the countable
property if every
contains
of the same size, where
within the space [
18]. Note that the
property moves away from the inclusion-ordering property of open sets in a topological space.
3. Chained Covering Spaces and Split–Joins
In this section, we present a set of definitions related to the formation of chained covering spaces and the algebraic split–join operations on the chains. The formulations of corresponding chain homotopies are also presented.
3.1. Chained Covering Spaces of n-Manifold Charts: Definitions
In this paper, the algebraic ordering relation is generally denoted as and we will represent a specific ordering relation chosen from the set as necessary to maintain clarity for specific cases. This is to emphasize that in the remaining sections of the paper the Riesz ordering relations in are preserved as when we are considering that the covering spaces are for the chart maps of a manifold (i.e., in the remainder of the paper we consider that and we write in place of for simplicity). The same goes for the linear ordering relation . Moreover, the algebraic chain in a topological space is denoted as and by following the conventions of homotopy theory of algebraic topology, we will denote the directed homotopy path in an algebraic chain as: (1) if the path follows monotonic sequence chosen in such that and (2) if the path follows monotonic sequence chosen in such that . For example, if an algebraic chain in a topological space is given by then and . Similarly, if the chain is given by then and . Furthermore, if and are two linear homotopy chain-paths in a topological space, then the homotopy path-composition is represented as by following the conventions of algebraic topology while preserving the order relation throughout the chain-homotopy.
Definition 10. Let be an n-manifold such that and be a chart map of the open neighborhood of generated by . The chart map is defined as algebraically chained if there is a finite ordered chain .
Observe that if the ordered chain is finite then in , where . In general, if we consider a topological space then is called a topological chained subspace. In an ordered chain in Hausdorff , any point is called a split-set. We can further extend this concept to the covering spaces which is defined as follows.
Definition 11. Let us consider a surjective covering map given by such that is a section at . The space is called as an algebraically (ordered) chained covering space of if contains a set of chains .
If we consider that is a specific chained covering space such that the set of chains is admitted, then it leads to the concept of twisted chained covering spaces, which is defined as follows.
Definition 12. Let us consider a covering map given by such that and the chart map is algebraically chained as . The covering map is called a twisted chained covering map if it admits the set of chains given as .
The schematic representation of chained-covering spaces of a chart
under foliation of an
n-manifold and the inclusion functions
and
is illustrated in
Figure 1. Note that the inclusions are not the Nachbin variety and the homeomorphism is an oriented isomorphism.
Remark 2. It is important to note that if there is an inclusion function such that is a finite chain preserving in then there are corresponding inclusion functions in covering spaces and preserving the properties that and , in the respective chained covering spaces.
3.2. Split–Join Combinatorial Chains: Definitions
Let us consider a general topological chained subspace , where the algebraic ordering relation is . If we choose any point then and are called the initial segment and final segment of if the following conditions are maintained.
Condition I: and,
Condition II:
We call the specific point a split–join point of the chain.
Remark 3. Every split–join point forms a split-set in a Hausdorff topological space.
Note that Condition II follows the concept of topological nets and if is uncountable with then the partitioning of a topological chained subspace has resemblance to the formation of basis elements of standard topology in . This preserves the Urysohn interpretations of real-valued continuous functions as chains. However, the multiple general topological chained subspaces contain a much richer combinatorial property of split–join chain set formation, which is defined as follows.
Definition 13. Let and be two topological chained subspaces such that and are finite as well as uncountable. If the function is a bijection then is called a split–join chain set at .
If we induce the algebraic ordering relation within the split–join algebraic chain set then it results in the formation of corresponding split–join homotopy chain-path, which is defined as follows.
Definition 14. A split–join algebraic chain set is called a split–join homotopy chain-path generated under relation , which is denoted as , if either or in .
Remark 4. It is important to note that, denotes that the split–join homotopy chain-path starts in and ends in . However, the split–join homotopy chain-path denotes that the path starts in chain and ends in chain .
3.3. Directions and Orientations
There are two possible combinatorial formations of split–join homotopy chain-paths within the topological spaces, which are called
simply directed homotopy chain-path and
oriented homotopy chain-path. A split–join homotopy chain-path
is called a
simply directed homotopy chain-path in a topological space if it preserves
any one of the following properties (either condition
or condition
) under bijective
, where either
or
in
.
This is to emphasize that any one of the aforesaid conditions needs to be satisfied by an algebraic chain-path in a topological space.
On the other hand, if we consider two topological chained subspaces
and
then
is called as an
oriented homotopy chain-path in a topological space under relation
if any one of the following conditions (either condition
or condition
) are satisfied.
This is to emphasize that any one condition within any one of the aforesaid sets of conditions need to be satisfied to form a single algebraic chain-path within a topological space. Moreover, the simply directed homotopy chain-paths under split–joins are formed when the chains are homogeneous in terms of algebraic relation. On the contrary, the oriented homotopy chain-paths under split–joins are formed when the algebraic relations are not homogeneous in two chains.
Example 1. As an example, we can represent two simply directed homotopy chain-paths derived from Equation (1) under split–join considering two chainsandretaining ordering relational homogeneity as: Similarly, the two oriented homotopy chain-paths can be derived from Equation (2) under split–join considering two chains
and
without preserving homogeneity of algebraic ordering relation as:
Geometrically, the algebraic split–join on a set of linearly ordered chains forms a variety of chain-paths under homotopic path-products as illustrated in
Figure 2.
The oriented homotopic path-products of chain-paths induce specific orientations of topological chained subspaces (paths) within the corresponding chained covering spaces, which is an additional characteristic property. However, the simply directed homotopy chain-paths linearly follow the algebraic ordering relation without showing any additional orientations of topological chain-paths within the spaces.
4. Main Results
The interplays between various types of algebraic chained covering spaces and chart maps of a manifold exhibit a rich set of topological properties. First, we show that the bijection between two algebraic chains in for some forms the simply directed homotopy chain-paths within the covering spaces.
Theorem 3. If is a covering space of for a manifold containing two algebraically ordered homogeneous chains and in then the bijection admits two simply directed homotopy chain-paths and at a split–join point .
Proof. Let
be an
n-manifold with a chart map
and
be a corresponding covering map. Suppose there are two algebraic chains
and
such that either
or
is maintaining homogeneity of algebraic ordering relation within
for some
. If we consider a bijection
then there is a split–join point
forming a set of sections of homotopy chain-paths within the covering spaces which are given as
where
and,
. It results in the formation of two split–join chain sets given as:
Thus, the corresponding two sets of simply directed homotopy chain-paths are admitted within
depending upon the algebraic ordering relation, which is given as follows.
Hence, the covering spaces admits two simply directed homotopy chain-paths and at a split–join point . □
Remark 5. It is easy to observe that a simply directed homotopy chain-path in the covering spaces maintains a path-homotopy property between the two split–join sets irrespective of the algebraic ordering relation. However, the two different simply directed homotopy chain-paths in the covering spaces are not in mutual path-homotopy. This results in the following corollary representing the topological properties of two homotopy chain-paths in view of algebraic topology.
Corollary 1. In a topological space two split–join homotopy chain-paths and do not mutually commute at any split–join point , where and .
Theorem 4. If is an algebraically chained covering map and is a twisted chained covering map of then the bijection forms two oriented homotopy chain-paths in .
Proof. Let
be a chart map of an n-manifold
. Suppose
and
are two covering maps such that
and
generates chained covering spaces, whereas
generates the corresponding twisted chained covering spaces. Accordingly, let us consider a bijection
, where
and
. If we choose a split–join point
, then by following the conditions of oriented homotopy chain-paths we can algebraically construct two homotopy chain-paths as follows.
Hence, the bijection forms two topologically oriented homotopy chain-paths in . □
Note that Theorem 3 and Theorem 4 give rise to a set of path-homotopies, which are algebraically chained topological subspaces in covering spaces under split–join operations. This set of path-homotopies of chain-paths is given as:
Moreover, the aforesaid theorem indicates that embedding of a homotopy chain-path in a depends on the nature of covering spaces and this observation leads to the following lemma.
Lemma 2. If an algebraically chained topological subspacecan be embedded intoby a continuous injective functionthen either the covering mapis an algebraic chain map or a twisted chain map, but not both.
Proof. The proof is relatively straightforward, and we prove it by following the method of contradiction. Suppose we consider a covering map which generates both the chained covering spaces and also the twisted covering spaces. Let us consider an algebraic chained topological subspace and a chain embedding under continuous injective function . In this case there are two covering sections such that . As the covering map is a surjection, thus if and then it leads to the conclusion that is not a chain and does not exist, which is a contradiction. Hence, the covering map can be either an algebraic chain map or a twisted chain map, but not both. □
Observe that the covering map is a variety of an evenly covering map of . We can further enrich the algebraic properties of the chained covering spaces of a chart map of . For example, the set of Hausdorff topological subspaces gives rise to a multiplicative group structure under the corresponding finite real-valued measure . This observation is presented in the following theorem.
Theorem 5. The set of Hausdorff topological subspaces forms a 5th order multiplicative group in chained covering spaces under the finite real-valued measure .
Proof. Let be a set of chained Hausdorff topological subspaces in the covering spaces containing split-sets and be a real-valued finite measure. Note that the condition is maintained by the measure. Suppose we fix considering the split-sets of cardinalities equal to 1. Hence, the measure forms a 5th order multiplicative group structure if , the real-valued measure preserves the condition , where . □
This immediately leads to the following lemma considering .
Lemma 3. If the cyclic groupis formed under the finite measure,then.
Proof. The proof is relatively straightforward. If is a 6th order cyclic group then admitting cyclic with generator . As a result, we can conclude that . □
Corollary 2. Everyin Hausdorffis finitely measurable under some real-valued measure.
Proof. Let us consider that is a Hausdorff topological space and the covering spaces are measure preserving. If in such that then the following condition will result: . It leads to two contradictions such as, (1) there are which are not locally compactable if and, (2) the generator cannot form a cyclic group algebraic structure if . Hence, we conclude that the measure is admitted. □