The countable chain condition and free algebras

1979 ◽  
Vol 165 (2) ◽  
pp. 101-106
Author(s):  
Stephen D. Comer
Keyword(s):  
Chain Condition ◽  
Free Algebras ◽  
Countable Chain Condition ◽  
Countable Chain

scholarly journals Lebesgue Measurability of Separately Continuous Functions and Separability

2007 ◽  
Vol 2007 ◽  
pp. 1-4
Author(s):  
V. V. Mykhaylyuk
Keyword(s):  
Topological Space ◽  
Chain Condition ◽  
Continuous Functions ◽  
Negative Answer ◽  
Regular Space ◽  
Property A ◽  
Completely Regular ◽  
Open Set ◽  
Countable Chain Condition ◽  
Countable Chain

A connection between the separability and the countable chain condition of spaces withL-property (a topological spaceXhasL-property if for every topological spaceY, separately continuous functionf:X×Y→ℝand open setI⊆ℝ,the setf−1(I)is anFσ-set) is studied. We show that every completely regular Baire space with theL-property and the countable chain condition is separable and constructs a nonseparable completely regular space with theL-property and the countable chain condition. This gives a negative answer to a question of M. Burke.

scholarly journals A characterization of the existence of a Souslin line

1982 ◽  
Vol 25 (3) ◽  
pp. 425-431
Author(s):  
Nobuyuki Kemoto
Keyword(s):  
Disjoint Union ◽  
Chain Condition ◽  
First Category ◽  
Countable Chain Condition ◽  
Isolated Points ◽  
Countable Chain

The main purpose of this paper is to show that there exists a Souslin line if and only if there exists a countable chain condition space which is not weak-separable but has a generic π-base. If I is the closure of the isolated points in a space X, then X is said to be weak-separable if a first category set is dense in X – I. A π-base is said to be generic if, whenever a member of is included in the disjoint union of members of it is included in one of them.

SETS WITH THE COUNTABLE CHAIN CONDITION

1987 ◽  
Vol 13 (1) ◽  
pp. 50
Author(s):  
Shortt
Keyword(s):  
Chain Condition ◽  
Countable Chain Condition ◽  
Countable Chain

A large power set axiom

1975 ◽  
Vol 40 (1) ◽  
pp. 48-54
Author(s):  
Paul E. Cohen
Keyword(s):  
Continuum Hypothesis ◽  
Chain Condition ◽  
Elementary Embedding ◽  
Large Power ◽  
Countable Chain Condition ◽  
Technical Lemma ◽  
Countable Chain ◽  
Suitable Treatment ◽  
Image Position ◽  
The Axiom Of Choice

Takeuti [6] has suggested the need for higher axioms for set theory which are analogous to higher axioms of infinity, but which claim that power sets are in some sense large. In this paper we investigate a reflection axiom of this sort (Axiom T).In §1, we introduce Axiom T and explore some related axioms. A technical lemma involving an elementary embedding is developed in §2 which allows us, in §3, to prove the relative consistency of Axiom T.The reader is assumed to be familiar with ramified forcing languages and the usual techniques of forcing. A suitable treatment of these subjects is given by Takeuti and Zaring [7].ZFC, GBC, ZF and GB are the set theories of Zermelo-Fraenkel and of Gödel-Bernays, with and without the axiom of choice (AC). CH is the continuum hypothesis.For α an ordinal, defineFor p1, p2 ∈ pα let p1 ≤ p2 mean that p1 ⊇ p2 (thus p1 is the stronger forcing condition). It is well known that Pα satisfies the countable chain condition [5].

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