Some independence results about compact metrizable spaces and two notions of finiteness

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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Universal elements for families of separable metrizable spaces

Journal of Mathematical Sciences ◽

10.1007/bf02434916 ◽

1998 ◽

Vol 91 (6) ◽

pp. 3387-3415

Author(s):

D. N. Georgiou ◽

S. D. Iliadis

Keyword(s):

Metrizable Spaces

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Independence of algebraic monodromy groups in compatible systems

Selecta Mathematica ◽

10.1007/s00029-021-00657-y ◽

2021 ◽

Vol 27 (4) ◽

Author(s):

Federico Amadio Guidi

Keyword(s):

Galois Representations ◽

Global Function ◽

Irreducible Decomposition ◽

Global Function Fields ◽

Normal Varieties ◽

Independence Result ◽

Monodromy Groups ◽

Independence Results ◽

Lisse Sheaves ◽

General Method

AbstractIn this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classes of automorphic forms, we prove the existence of a Lie-irreducible decomposition. From this we deduce an independence result. We conclude with the case of compatible systems of Galois representations over global function fields, for which we prove the existence of a Lie-irreducible decomposition, and we deduce an independence result. From this we also deduce an independence result for compatible systems of lisse sheaves on normal varieties over finite fields.

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Further independence results

Lecture Notes in Mathematics - Proper Forcing ◽

10.1007/bfb0096600 ◽

1982 ◽

pp. 388-393

Keyword(s):

Independence Results

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Algebraic Independence Results Related to 〈q, r〉-Number Systems

Monatshefte für Mathematik ◽

10.1007/s00605-005-0320-5 ◽

2006 ◽

Vol 147 (4) ◽

pp. 319-335 ◽

Cited By ~ 3

Author(s):

Shin-ichiro Okada ◽

Iekata Shiokawa

Keyword(s):

Algebraic Independence ◽

Number Systems ◽

Independence Results

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Optimal metrics on metrizable spaces

Archiv der Mathematik ◽

10.1007/bf01222632 ◽

1971 ◽

Vol 22 (1) ◽

pp. 660-663

Author(s):

Ludvik Janos

Keyword(s):

Metrizable Spaces

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Field Independence and Resistance to Reversal of Perspective

Perceptual and Motor Skills ◽

10.2466/pms.1966.22.2.543 ◽

1966 ◽

Vol 22 (2) ◽

pp. 543-546 ◽

Cited By ~ 20

Author(s):

Frank Haronian ◽

A. Arthur Sugerman

Keyword(s):

Short Form ◽

Necker Cube ◽

Normal Male ◽

Field Independence ◽

Embedded Figures Test ◽

Following Instructions ◽

Independence Results ◽

Male University Students ◽

Rod And Frame Test ◽

Rod And Frame

The more successful 102 normal male university students were in following instructions to resist fluctuations of the Necker cube, the more field-independently they scored on both Series III of the rod-and-frame test ( r = .28) and on Jackson's short form of the embedded-figures test ( r = .24). Under neutral instructions, the correlations were nil. Results support prior findings that a small but significant portion of the variance of Necker cube fluctuations under instructions to control the rate of shift is related to scores of field independence. Results support Jackson's finding that ability to control the rate of shift is not related to intelligence.

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