Partitioning Topological Spaces

Weiss, William

doi:10.1007/978-3-642-72905-8_11

Cited by 21 publications

(12 citation statements)

References 9 publications

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“…Meanwhile, both ω 1 → (ω 1 , α) 2 for all α ∈ ω 1 [Tod83] and ω 1 → (ω 1 , ω + 2) 2 [Haj60] are consistent with ZFC, though the topological version of the former remains unchecked. Finally, it is also known that β → top (ω + 1) 2 ℵ0 for all ordinals β [Wei90]. When n > 2, not much more can be said in the setting of countable ordinals, since Kruse showed that if n ≥ 3 and β is a countable ordinal, then β → (ω + 1, n + 1) n [Kru65].…”

Section: Introductionmentioning

confidence: 99%

Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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, on the occasion of his birthday.Abstract. We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write β →top (α, k) 2 to mean that, for every red-blue coloring of the collection of 2-sized subsets of β, there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k. The least such β is the topological Ramsey number R top (α, k).We prove a topological version of the Erdős-Milner theorem, namely that R top (α, k) is countable whenever α is countable. More precisely, we prove that R top (ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and finite k. Our proof is modeled on a new easy proof of a weak version of the Erdős-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of α, proving among other results thatOur computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.

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Section: Introductionmentioning

confidence: 99%

Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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“…The space partition Π X = {A i ⊂ X : i ∈ I} is a family of subspaces such that, ∀A i , A j ∈ Π X , A i ∩ A j = φ and, ∪ ∀i∈I A i = X. The partitioned topological spaces may be employed to obtain a homogeneous set, which is topologically relevant in nature [27].…”

Section: Topological Space Partitionmentioning

confidence: 99%

Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings

Bagchi

2020

Symmetry

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In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

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“…A homogeneous set can be derived from a partitioned topological space [25]. The partitioning of the topological space based on the generation of a quotient set containing equivalence classes may not affect separability of the original space.…”

Section: Embeddable Topological Decomposition Of Groupsmentioning

confidence: 99%

The Sequential and Contractible Topological Embeddings of Functional Groups

Bagchi

2020

Symmetry

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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.

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Partitioning Topological Spaces

Cited by 21 publications

References 9 publications

Topological Ramsey numbers and countable ordinals

Topological Ramsey numbers and countable ordinals

Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings

The Sequential and Contractible Topological Embeddings of Functional Groups

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