Affine Structure in 3-Manifolds: VI. Compact Spaces Covered by Two Euclidean Neighborhoods

In this paper soft pre separations are defined and few of their properties are stated and proved. Also soft pre compactness is defined and properties of a such a space are discussed.

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On 𝒞ℛ-Epic Embeddings and Absolute 𝒞ℛ-Epic Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2005-043-2 ◽

2005 ◽

Vol 57 (6) ◽

pp. 1121-1138 ◽

Cited By ~ 1

Author(s):

Michael Barr ◽

R. Raphael ◽

R. G. Woods

Keyword(s):

Locally Compact ◽

Compact Spaces ◽

Open Questions ◽

Tychonoff Spaces ◽

Locally Compact Spaces

AbstractWe study Tychonoff spaces X with the property that, for all topological embeddings X → Y, the induced map C(Y ) → C(X) is an epimorphism of rings. Such spaces are called absolute 𝒞ℛ-epic. The simplest examples of absolute 𝒞ℛ-epic spaces are σ-compact locally compact spaces and Lindelöf P-spaces. We show that absolute CR-epic first countable spaces must be locally compact.However, a “bad” class of absolute CR-epic spaces is exhibited whose pathology settles, in the negative, a number of open questions. Spaces which are not absolute CR-epic abound, and some are presented.

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Quantum Orlicz Spaces in Information Geometry

Open Systems & Information Dynamics ◽

10.1007/s11080-004-6626-2 ◽

2004 ◽

Vol 11 (04) ◽

pp. 359-375 ◽

Cited By ~ 13

Author(s):

R. F. Streater

Keyword(s):

Quantum Information ◽

Tangent Space ◽

Selfadjoint Operator ◽

Affine Connection ◽

Orlicz Spaces ◽

Trace Class ◽

Information Geometry ◽

Young Function ◽

Affine Structure ◽

Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

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nIαg-Compact spaces and nIαg-Lindelof spaces in nano ideal topological spaces

10.1063/5.0025274 ◽

2020 ◽

Author(s):

M. Parimala ◽

D. Arivuoli ◽

R. Perumal ◽

S. Krithika

Keyword(s):

Topological Spaces ◽

Compact Spaces

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On the Dimension Modulo Classes of Topological Spaces and Free Topological Groups

Georgian Mathematical Journal ◽

10.1515/gmj.2003.209 ◽

2003 ◽

Vol 10 (2) ◽

pp. 209-222

Author(s):

I. Bakhia

Keyword(s):

Wide Class ◽

Sufficient Conditions ◽

Topological Spaces ◽

Topological Groups ◽

Compact Spaces ◽

Topological Semigroups

Abstract Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

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Domination by countably compact spaces and hyperspaces

Topology and its Applications ◽

10.1016/j.topol.2018.12.001 ◽

2019 ◽

Vol 253 ◽

pp. 38-47 ◽

Cited By ~ 1

Author(s):

Daniel Jardón

Keyword(s):

Compact Spaces ◽

Countably Compact ◽

Countably Compact Spaces

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Several results on compact metrizable spaces in $$\mathbf {ZF}$$

Monatshefte für Mathematik ◽

10.1007/s00605-021-01582-0 ◽

2021 ◽

Author(s):

Kyriakos Keremedis ◽

Eleftherios Tachtsis ◽

Eliza Wajch

Keyword(s):

Continuum Hypothesis ◽

Compact Spaces ◽

Power Set ◽

Compact Metrizable Space ◽

Permutation Models ◽

Independence Results ◽

The Axiom Of Choice ◽

Metrizable Spaces ◽

Transfer Theorems ◽

The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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