2. Preliminary Concepts
In this section, a set of basic concepts and definitions are presented to establish preliminary notions about topological embeddings, groups and related analysis. In this paper,
denotes the closure of the corresponding open set
, and the complement of set
is given by
. Furthermore, it is considered that the normal topological space has a Hausdorff closure property, which may contain fixed points [
10]. The power set of any arbitrary set
is denoted by
. The symbol
denotes that,
is a subgroup of
.
Let
be a set and
be a topological space, which is considered to be Hausdorff to avoid multiconvergence localities. In a normal and complete Hausdorff topological space, the bijection
contains a set of fixed points given by,
. Let
be equipped with abstract algebraic operation
such that the operation is closed in the set. The structure
is called a group if it maintains the properties [
9,
21], (1)
, (2)
and, (3)
. The element
is unique and
indicates that
.
A group
can be equipped with a topological structure. The topology on
is denoted by
, where it maintains the axioms of topology. A group action is given by
under specific conditions, where
is a set on which a group is acted on. The required properties related to group action are (i)
, and (ii)
. The second property asserts associativity of
in the presence of group action. The identity function is given by
such that
in any arbitrary set
. If
is a groupoid, then the morphism is given by
in the groupoid [
22]. The morphisms
maintain associative composition law given by
.
A topological group is a variety of groups in a topological space such that continuity of space under the closed algebraic operation within the space is maintained. The structure
is a topological group if it maintains the axioms [
23], (a)
is a group, (b)
is a topological space, and (c) the algebraic operation
is continuous along with the continuous existence of
. Furthermore, if
in
are two elements and
is the fixed element, then the transformation
is a homeomorphism of
into itself. A topological group
can be compactly generated if
and
such that
is a subgroup in compact subspace [
9].
The Jordan curve
in a topological space
representing a plane, is a function
, where
is a closed curve in 2-D real such that
is continuous and injective [
14]. It is to note that
can be any polygon in space and
homeomorphic to
can be considered as a circle group. There can be embeddings of closed curves in surfaces leading to the Jordan curve theorem (JCT) [
24]. The embedding of
in a surface
is called 2-cell embedding if all the regions are 2-cells [
15]. Interestingly, in group algebra, the semigroups can be embedded within the group structures, which is actually not a topological embedding. However, the formulation of embedding in a group is not a straight-forward approach. The Steinitz embedding theorem shows that every integral domain can be embedded in a field [
16]. The Steinitz theorem relies on the construction of ordered pairs of elements maintaining an equivalence relation.
4. Main Results
This section presents the main results as a set of theorems derived from the proposed concepts, definitions and structures. The properties of functional group embeddings in normal topological spaces are formulated considering that the embeddings are homeomorphic to (i.e., they are complete with one-point compactification). Moreover, the embeddings of functional groups into a normal topological space are sequential in nature. In a stricter sense, the condition of formation of a monotone class is considered as a special case.
In the beginning, it is shown that, the sequences of embeddings are mutually disjoint within the normal topological space.
Theorem 1. Inof anormal topological space, ifand, thensuch that.
Proof. Let be a topological space and be a finite sequence of functional groups. Let be the embedding in corresponding open set , where . If is a finite sequence of embeddings in , then the sequence of components will be generated in . Let us consider that , and in . This indicates that if is an embedding with , then , where is open in such that . Thus, considering that is a normal topological space, if is such that , then is a monotone class in if and only if whenever . Hence, in open subspace, it is true that and in . Moreover, as and are two functional groups, so as they are distinct and separated. Hence, in the normal topological space and , where . □
If one considers that a monotone class of group embeddings in a topological space is mutually disjoint, then there exists a sequence of fixed points in the set of generated components. If the embedding space is finite, then the sequence is also finite.
Theorem 2. Ifis a sequence of disjoint functional group embeddings in finite normal topological space, thenis a finite sequence of fixed points in.
Proof. Let be a normal topological space having Hausdorff property. If is finite, then such that is open and finite indicating that is also finite (but not necessarily compact). Let be a sequence of functional group embeddings in respective such that whenever and . Let be a sequence of Jordan components generated by in the corresponding . If is a monotone class such that , then in the topological space , where if and . Let be a pair-wise continuous function in such that , where . As , so is open in indicating that the function is bounded in every . Thus, such that . Hence, the is a sequence of fixed points in normal topological space under disjoint functional group embeddings. □
Lemma 1. Ifis compact, thenis convergent, where is a limit.
Proof. Let be a compact topological space and be a sequence of embeddings of functional groups in into open subspace . If and generate two closed components in , then and respectively, where and . However, as in because the functional groups are mutually disjoint, so in . Moreover, open sets in normal topological space such that and whenever . Thus, sequence of embeddings generates in . Again, if is compact, then is compact, where in . Thus, in the normal and closed topological subspace, , where . Hence, if is a sequence of fixed-points in compact , then and such that in . Furthermore, as so in . □
It is noted earlier that the embeddings of functional groups are homeomorphic to in topological space. The neighborhood system of fixed points of the convergent sequence in embedding subspace characterizes the nature of embeddings. It also reaffirms the condition that the underlying topological space is normal.
Theorem 3. Ifandare functional group embeddings in normal topological space, then the fixed pointsandhave neighborhoods such that, whereand.
Proof. Let and be two functional groups such that . Let the two corresponding embeddings be and in the normal topological space . If is a convergent sequence in and , then such that , where and are fixed points in the respective closed components in . However, implication is maintained in open subspace and as , so maintaining disjoint embedding condition. Moreover, the embeddings are homeomorphic to and generate closed components in . Thus, it is indeed true that, , and in the topological space . As the topological space is normal as well as Hausdorff, hence and such that , where in normal . □
Remark 2. The extension of the above property indicates that normal topological space allows normal embedded subspaces. As the topological spaceis normal, thussuch thatand. Moreover, the normal subspace containing the monotone class embeddings maintains the condition given as, andin.
Interestingly, the mutual disjoint embedding of functional groups is independent of order relation in embedding sequence. This property is maintained as long as a monotone class is formed within a topological subspace. It is important to note that, when a converging sequence of fixed points are considered within a compact subspace containing embeddings, the order of embeddings is considered to be fixed according to the sequence.
Theorem 4. In the normal topological space, ifsuch that, then the components are not separable independent of embedding sequence.
Proof. Let be a normal topological space and be in the subspace , where the subspace is open and . Suppose, the components are separable in independent of embeddings sequence in . Thus, if in , then open subspaces such that and . Thus, and are separations in . However, in this case, either or , which is a contradiction. Thus, if is in the embedding subspace , then and , where . Hence, the closure is not a separable subspace and as a result the components are not separable or independent of embedding sequence. □
Corollary 1. The uniform contraction has a role in determining separation in topological spaces. In the normal topological space, ifsuch thatandthen. This is a relatively straight conclusion from the above-mentioned theorem.
The continuous contraction in normal topological spaces homeomorphic to invites the requirement of surjectivity in a sequence of embeddings. However, such a sequence should form a monotone class in a normal topological subspace.
Theorem 5. In a normal topological space, the uniform contractionis a surjection in, where.
Proof. Let be a normal topological space and such that , where . Let be a uniform contraction and . Now, the components and are dense because and in , where are functional group embeddings in topological subspace. Moreover, as is not separable and , thus are not separable. Again as in the sequence of embeddings in the normal subspace . Moreover, the sequence of embeddings maintains the homeomorphism as, in . Thus, if is a continuous contraction, then such that , where are open neighborhoods of respectively. Hence, the uniform contraction is a surjection in , where the topological space is normal in nature. □
The surjective contraction supports the monotone class of components generated by a sequence of functional group embeddings in a normal topological subspace. The existence of Kakutani fixed points in a converging sequence in such a subspace containing embedded components invites the semicontinuity of finite set valued function within the normal subspace containing embeddings sequence. This property is presented in the next theorem.
Theorem 6. Ifandwithare two Kakutani fixed points on embeddings in a normal topological space, then there is a semicontinuous set-valued functionsuch that, whereandis an open set.
Proof. Let be a normal topological space having embedded sequence of functional groups forming a monotone class structure generated by . Let be such that there are two Kakutani fixed points and with in . If is a set-valued function, then such that , where are open sets in . However, if is semicontinuous, then indicates that the mapping is unique. Moreover, in the normal subspace of embedding, and the sequence of fixed points in is a converging sequence, where . Thus, open set such that . If then it leads to the conclusion that, . Otherwise, if then such that , where . Hence, in any case, such that in . □
Generally, Schoenflies homeomorphism is defined in between two separable topological spaces containing embeddings. Thus, the Schoenflies homeomorphism can exist within multiple subspaces in a normal topological space. Suppose multiple functional group embeddings are distributed within the subspaces of a normal topological space. If the functional groups are disjoint and homeomorphic, then there can be interplay between embedded group homeomorphism and Schoenflies homeomorphism. This interaction is presented as a theorem next.
Theorem 7. Letbe a normal topological space andbe such that. If Schoenflies homeomorphism exists as, then there is a group homeomorphism between embedded functional groupsandinand, respectively.
Proof. Let be a normal topological space and the two disjoint subspaces are, . Let and be two functional groups such that and . Thus, there can be two functional group embeddings in normal given as, and in the respective subspaces, which are normal category of topological spaces. If the Schoenflies homeomorphism exists in , then . Thus, in , such that in . Moreover, as is a functional group, hence is maintained due to closure property of the embedded functional group. Let be a bijection in . If one considers that by following existing Schoenflies homeomorphism, then such that in the embedded subspace, where . Hence, is an embedded functional group homeomorphism preserved by . □
Remark 3. From the above theorem, it can be further concluded thatin the normal topological spaceand the embeddings are not contractible in nature. As a result, the sequences of homeomorphic functional group embeddings are not in a monotone class.
The above observation can be extended under uniform contraction between two homeomorphic embedding spaces, which are separable. The group homeomorphism exists under uniform contraction if and only if the embedding spaces maintain Schoenflies homeomorphism in separable subspaces as given in the next theorem.
Theorem 8. Letbe a normal topological space andbe such that, Ifandare embeddings in respective subspaces, such thatandare group homeomorphisms, then there are uniform contractionsandif Schoenflies homeomorphism exists as,.
Proof. Let be a normal topological space and be such that and be a set of disjoint functional groups. Let and be the respective embeddings in corresponding subspaces such that and in . Let be a Schoenflies homeomorphism such that and . Thus, the functional group embeddings and Schoenflies homeomorphism will generate components in subspaces as and . If is a uniform contraction, then such that and . Suppose and are embedded functional groups’ homeomorphisms in normal topological space, . Thus, within the homeomorphic group embedding subspace, and , where are respective functional groups. Similarly, and , where are respective functional groups and . Hence, there exists a uniform contraction, in such that such that and , where and . □
This indicates that there is a strong isometry between two separable normal topological subspaces containing functional group embeddings if the disjoint embedded functional groups are homeomorphic to each other. Moreover, in this case, the Schoenflies homeomorphism exists between the subspaces.