Weakly Semialgebraic Spaces

Knebusch, Manfred

doi:10.1007/bfb0084987

Cited by 14 publications

(76 citation statements)

References 8 publications

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“…A weakly definable space (over R) is a space M (over R) having a family, indexed by a partially ordered set A, of regular closed definable subsets (M α ) α∈A such that the following conditions hold: WD1) M is the union of all M α , WD2) if α ≤ β then M α is a (closed) subspace of M β , WD3) for each α there is only a finite number of β such that β ≤ α, WD4) the family (M α ) is strongly inverse directed, i. e. for each α, β there is some γ such that γ ≤ α, γ ≤ β and M γ = M α ∩ M β , WD5) the set of indices is directed: for each α, β there is γ with γ ≥ α, γ ≥ β, WD6) the space M is the inductive limit of the spaces (M α ), what means the following: A weakly definable subset is such a subset X ⊆ M that: has definable intersections with all members of some exhaustion (M α ), and is considered with the exhaustion (X ∩ M α ), hence it may be considered as a subspace of M (cf. IV.3 in [19]).…”

Section: Weakly Definable Spacesmentioning

confidence: 95%

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O-minimal homotopy and generalized (co)homology

Piękosz¹

2013

Rocky Mountain J. Math.

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This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspond (up to an isomorphism) to the known topological generalized homology and cohomology theories on pointed CW-complexes.2000 Mathematics Subject Classification: 03C64, 55N20, 55Q05.

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Section: Weakly Definable Spacesmentioning

confidence: 95%

“…The intention of the author of the present paper is not to re-write about 600 pages with this simple change, but to give enough understanding of the theory to the reader. Some examples and facts from [9] and [19] are restated to make this understanding easy. (The above remarks apply to so-called "geometric" theory.…”

Section: Introductionmentioning

confidence: 99%

O-minimal homotopy and generalized (co)homology

Piękosz¹

2013

Rocky Mountain J. Math.

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“…Since there does not exist A ∈ B such that J ⊆ int τ (ρ) A, we deduce from Theorem 1.9 that the gts (X, EF(τ (ρ), B)) is not Sm-quasi-metrizable. The space (X, τ (ρ)) can be called the comb with its hand J and teeth J q , q ∈ Q (compare with Example IV.4.7 of [14]).…”

Section: Examplementioning

confidence: 99%

“…The original definition of a generalized topological space given in [5] was simplified in [17]. It was shown in [14], [18] and [19] that in the context of locally definable spaces it is sufficient to use function sheaves instead of general sheaves. Our direct approach to the investigation of Delfs-Knebusch gtses seems simpler and more natural for problems of general topology than that of [6].…”

Section: Introductionmentioning

confidence: 99%

Bornological quasi-metrizability in generalized topology

Piękosz¹,

Wajch²

2018

HJMS

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A concept of quasi-metrizability with respect to a bornology of a generalized topological space in the sense of Delfs and Knebusch is introduced. Quasi-metrization theorems for generalized bornological universes are deduced. A uniform quasi-metrizability with respect to a bornology is studied. The class of locally small spaces is considered and a possibly larger class of weakly locally small spaces is defined. The proofs and numerous examples are given in ZF. An example of a weakly locally small space which is not locally small is constructed under ZF+CC. Several categories, relevant to generalized bornological universes, are defined and shown to be topological constructs.

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“…[11]; [27]) to cover the geometry of arbitrary real spectra. But there are other contexts where these rings appear naturally.…”

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confidence: 99%

Gabriel filters in real closed rings

Schwartz

1997

Commentarii Mathematici Helvetici

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Abstract. Real closed rings arise in semi-algebraic geometry and topology as well as in the investigation of partially ordered rings. It is shown that localizations of real closed rings with respect to Gabriel filters, or more generally: multiplicative filters, are again real closed. Thus, real closedness is preserved under a large number of important ring theoretic constructions. For a few particularly simple cases the multiplicative filters are classified and the localizations are determined.Mathematics Subject Classification (1991). 13B30, 14P10, 06F25, 54C30.

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Weakly Semialgebraic Spaces

Cited by 14 publications

References 8 publications

O-minimal homotopy and generalized (co)homology

O-minimal homotopy and generalized (co)homology

Bornological quasi-metrizability in generalized topology

Gabriel filters in real closed rings

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