Abstract
The continuous and injective embeddings of closed curves in Hausdorff topological spaces maintain isometry in subspaces generating components. An embedding of a circle group within a topological space creates isometric subspace with rotational symmetry. This paper introduces the generalized algebraic construction of functional groups and its topological embeddings into normal spaces maintaining homeomorphism of functional groups. The proposed algebraic construction of functional groups maintains homeomorphism to rotationally symmetric circle groups. The embeddings of functional groups are constructed in a sequence in the normal topological spaces. First, the topological decomposition and associated embeddings of a generalized group algebraic structure in the lower dimensional space is presented. It is shown that the one-point compactification property of topological space containing the decomposed group embeddings can be identified. Second, the sequential topological embeddings of functional groups are formulated. The proposed sequential embeddings follow Schoenflies property within the normal topological space. The preservation of homeomorphism between disjoint functional group embeddings under Banach-type contraction is analyzed taking into consideration that the underlying topological space is Hausdorff and the embeddings are in a monotone class. It is shown that components in a monotone class of isometry are not separable, whereas the multiple disjoint monotone class of embeddings are separable. A comparative analysis of the proposed concepts and formulations with respect to the existing structures is included in the paper.
Highlights
The continuous and injective embeddings in topological spaces have wide varieties as well as respective applications
Banach-type contraction is analyzed taking into consideration that the underlying topological space is Hausdorff and the embeddings are in a monotone class
The properties of functional group embeddings in normal topological spaces are formulated considering that the embeddings are homeomorphic to S1
Summary
The continuous and injective embeddings in topological spaces have wide varieties as well as respective applications. The topological embeddings consider connected spaces, which are at least first countable in nature. The various categories of topological embeddings and associated homeomorphisms give rise to a set of interesting and insightful properties. The topological embeddings of decomposed groups have a wide array of applications [3,4]. The topological embeddings of decomposed structures can be placed in two categories. This paper proposes the generalized algebraic formulation of functional groups and their sequential, as well as contractible embeddings in normal topological spaces. A brief description of various aspects of topological embeddings is presented in order to establish elementary concepts. The research questions dealt in this paper and the motivational aspects are presented
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