On the role of the Baire Category Theorem and Dependent Choice in the foundations of logic

1985 ◽  
Vol 50 (2) ◽  
pp. 412-422 ◽  
Author(s):  
Robert Goldblatt
Keyword(s):  
Existence Theorem ◽  
Metric Spaces ◽  
Baire Category ◽  
Baire Category Theorem ◽  
Hausdorff Spaces ◽  
Complete Metric Spaces ◽  
Category Theorem ◽  
Dependent Choice

AbstractThe Principle of Dependent Choice is shown to be equivalent to: the Baire Category Theorem for Čech-complete spaces (or for complete metric spaces); the existence theorem for generic sets of forcing conditions; and a proof-theoretic principle that abstracts the “Henkin method” of proving deductive completeness of logical systems. The Rasiowa-Sikorski Lemma is shown to be equivalent to the conjunction of the Ultrafilter Theorem and the Baire Category Theorem for compact Hausdorff spaces.

A Note on Small Baire Spaces

1986 ◽  
Vol 38 (3) ◽  
pp. 659-665 ◽  
Author(s):  
Saharon Shelah ◽  
Stevo Todorcevic
Keyword(s):  
Topological Space ◽  
Metric Spaces ◽  
Baire Category ◽  
Baire Space ◽  
Natural Question ◽  
Baire Category Theorem ◽  
Complete Metric Spaces ◽  
Compact Spaces ◽  
Category Theorem ◽  
Regular Open

A Baire space is a topological space which satisfies the Baire Category Theorem, i.e., in which the intersection of countably many dense open sets is dense. In this note we shall be interested in the size of Baire spaces, so to avoid trivialities we shall consider only non-atomic spaces, that is, spaces X whose regular open algebras ro(X) are non-atomic. All natural examples of Baire spaces, such as complete metric spaces or compact spaces, seem to have sizes at least 2ℵ0. So a natural question, asked first by W. Fleissner and K. Kunen [5], is whether there exists a Baire space of the minimal possible size ℵ1. The purpose of this note is to show that such a space need not exist by proving the following result.

scholarly journals Some remarks on fuzzy k-pseudometric spaces

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3567-3580 ◽  
Author(s):  
Alexander Sostak
Keyword(s):  
Metric Spaces ◽  
Baire Category ◽  
Important Class ◽  
Baire Category Theorem ◽  
Category Theorem ◽  
Type Spaces ◽  
Fixed Constant

An important class of spaces was introduced by I.A. Bakhtin (under the name ?metric-type?) and independently rediscovered by S. Czerwik (under the name ?b-metric?). Metric-type spaces generalize ?classic? metric spaces by replacing the triangularity axiom with a more general axiom d(x,z)? k? (d(x,y)+ d(y,z)) for all x,y,z ? X where k ? 1 is a fixed constant. Recently R. Saadadi has introduced the fuzzy version of ?metric-type? spaces. In this paper we consider topological and sequential properties of such spaces, illustrate them by several examples and prove a certain version of the Baire Category Theorem.

scholarly journals A note on Baire spaces and continuous lattices

1980 ◽  
Vol 21 (2) ◽  
pp. 265-279 ◽  
Author(s):  
Karl H. Hofmann
Keyword(s):  
Spectral Theory ◽  
Baire Category ◽  
Baire Category Theorem ◽  
Hausdorff Spaces ◽  
Theoretical Methods ◽  
Category Theorem ◽  
Continuous Lattices ◽  
Non Hausdorff Spaces

We prove a Baire category theorem for continuous lattices and derive category theorems for non-Hausdorff spaces which imply a category theorem of Isbell's and have applications to the spectral theory of C*-algebras. The same lattice theoretical methods yield a proof of de Groot's category theorem for regular subcompact spaces.

scholarly journals The closed graph theorem without category

1987 ◽  
Vol 36 (2) ◽  
pp. 283-287 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Baire Category ◽  
Baire Category Theorem ◽  
Normed Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Category Theorem

We show that a diagonal theorem of P. Antosik can be used to give a proof of the Closed Graph Theorem for normed spaces which does not depend upon the Baire Category Theorem.

The Baire category theorem in weak subsystems of second-order arithmetic

1993 ◽  
Vol 58 (2) ◽  
pp. 557-578 ◽  
Author(s):  
Douglas K. Brown ◽  
Stephen G. Simpson
Keyword(s):  
Functional Analysis ◽  
Model Theory ◽  
Baire Category ◽  
Second Order ◽  
Base System ◽  
Baire Category Theorem ◽  
Second Order Arithmetic ◽  
Category Theorem ◽  
Order Arithmetic

AbstractWorking within weak subsystems of second-order arithmetic Z2 we consider two versions of the Baire Category theorem which are not equivalent over the base system RCA0. We show that one version (B.C.T.I) is provable in RCA0 while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of Z2, which we call and , and , show that suffices to prove B.C.T.II. Some model theory of and its importance in view of Hilbert's program is discussed, as well as applications of our results to functional analysis.

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