A Corson compact L-space from a Suslin tree

2015 ◽  
Vol 141 (2) ◽  
pp. 149-155
Author(s):  
Peter Nyikos
Keyword(s):  
Suslin Tree ◽  
Corson Compact

On Σ1 well-orderings of the universe

1976 ◽  
Vol 41 (1) ◽  
pp. 167-170
Author(s):  
Leo Harrington ◽  
Thomas Jech
Keyword(s):  
Simple Formula ◽  
Generic Extension ◽  
Joint Paper ◽  
Suslin Tree ◽  
The Universe

The constructible universe L of Gödel [2] has a natural well-ordering <L given by the order of construction; a closer look reveals that this well-ordering is definable by a Σ1 formula. Cohen's method of forcing provides several examples of models of ZF + V ≠ L which have a definable well-ordering but none is definable by a relatively simple formula.Recently, Mansfield [7] has shown that if a set of reals (or hereditarily countable sets) has a Σ1, well-ordering then each of its elements is constructible. A question has thus arisen whether one can find a model of ZF + V ≠ L that has a Σ1 well-ordering of the universe. We answer this question in the affirmative.The main result of this paper isTheorem. There is a model of ZF + V ≠ L which has a Σ1 well-ordering.The model is a generic extension of L by adjoining a branch through a Suslin tree with certain properties. The branch is a nonconstructible subset of ℵ1. Note that by Mansfield's theorem, the model must not have nonconstructible subsets of ω.Our results can be generalized in several directions. We note that in particular, we can get a model with a Σ1 well-ordering that is not L[X] for any set X. As one might expect from a joint paper by a recursion theorist and a set theorist, the proof consists of a construction and a computation.

A Corson Compact Space is Countable if the Complement of its Diagonal is Functionally Countable

2021 ◽  
Vol 58 (3) ◽  
pp. 398-407
Author(s):  
Vladimir V. Tkachuk
Keyword(s):  
Continuous Function ◽  
Compact Space ◽  
Infinite Cardinal ◽  
Scattered Space ◽  
Corson Compact

A space X is called functionally countable if ƒ (X) is countable for any continuous function ƒ : X → Ø. Given an infinite cardinal k, we prove that a compact scattered space K with d(K) > k must have a convergent k+-sequence. This result implies that a Corson compact space K is countable if the space (K × K) \ ΔK is functionally countable; here ΔK = {(x, x): x ϵ K} is the diagonal of K. We also establish that, under Jensen’s Axiom ♦, there exists a compact hereditarily separable non-metrizable compact space X such that (X × X) \ ΔX is functionally countable and show in ZFC that there exists a non-separable σ-compact space X such that (X × X) \ ΔX is functionally countable.

Jensen's ⃞ principles and the Novák number of partially ordered sets

1986 ◽  
Vol 51 (1) ◽  
pp. 47-58 ◽  
Author(s):  
Boban Veličković
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Large Cardinals ◽  
Ordered Sets ◽  
Weakly Compact ◽  
Cardinal Function ◽  
Partially Ordered ◽  
Suslin Tree

In this paper we consider various properties of Jensen's □ principles and use them to construct several examples concerning the so-called Novák number of partially ordered sets.In §1 we give the relevant definitions and review some facts about □ principles. Apart from some simple observations most of the results in this section are known.In §2 we consider the Novák number of partially ordered sets and, using □ principles, give counterexamples to the productivity of this cardinal function. We also formulate a principle, show by forcing that it is consistent and use it to construct an ℵ2-Suslin tree T such that forcing with T × T collapses ℵ1.In §3 we briefly consider games played on partially ordered sets and relate them to the problems of the previous section. Using a version of □ we give an example of a proper partial order such that the game of length ω played on is undetermined.In §4 we raise the question of whether the Novák number of a homogenous partial order can be singular, and show that in some cases the answer is no.We assume familiarity with the basic techniques of forcing. In §1 some facts about large cardinals (e.g. weakly compact cardinals are -indescribable) and elementary properties of the constructible hierarchy are used. For this and all undefined terms we refer the reader to Jech [10].

scholarly journals Functional-analytic properties of Corson-compact spaces

1988 ◽  
Vol 89 (3) ◽  
pp. 197-229 ◽  
Author(s):  
S. Argyros ◽  
S. Mercourakis ◽  
S. Negrepontis
Keyword(s):  
Compact Spaces ◽  
Corson Compact

scholarly journals CHAINS OF FUNCTIONS IN -SPACES

2015 ◽  
Vol 99 (3) ◽  
pp. 350-363 ◽  
Author(s):  
TOMASZ KANIA ◽  
RICHARD J. SMITH
Keyword(s):  
Linear Operators ◽  
Partial Ordering ◽  
Hausdorff Spaces ◽  
Weakly Compact ◽  
Compact Spaces ◽  
Classical Work ◽  
Ordered Space ◽  
Weakly Compact Operators ◽  
Topology Of Pointwise Convergence ◽  
Corson Compact

The Bishop property (♗), introduced recently by K. P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński’s classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of (♗): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after (♗) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no nonmetrizable linearly ordered space, then every member of $\mathscr{D}$ has (♗). Examples of such classes include all $K$ for which $C(K)$ is Lindelöf in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying (♗) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

PRESERVATION OF SUSLIN TREES AND SIDE CONDITIONS

2016 ◽  
Vol 81 (2) ◽  
pp. 483-492
Author(s):  
GIORGIO VENTURI
Keyword(s):  
Countable Support ◽  
Suslin Tree ◽  
Preservation Theorem ◽  
Proper Forcing ◽  
Countable Support Iteration ◽  
Forcing Axiom

AbstractWe show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Suslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known using a standard countable support iteration and a preservation theorem due to Miyamoto.

scholarly journals Every Radon-Nikodym Corson compact space is Eberlein compact

1991 ◽  
Vol 98 (2) ◽  
pp. 157-174 ◽  
Author(s):  
J. Orihuela ◽  
W. Schachermayer ◽  
M. Valdivia
Keyword(s):  
Compact Space ◽  
Eberlein Compact ◽  
Corson Compact

scholarly journals Forcings constructed along morasses

2011 ◽  
Vol 76 (4) ◽  
pp. 1097-1125 ◽  
Author(s):  
Bernhard Irrgang
Keyword(s):  
Hausdorff Topology ◽  
Simple Observation ◽  
Alternative Construction ◽  
Suslin Tree ◽  
A Chain

AbstractWe further develop a previously introduced method of constructing forcing notions with the help of morasses. There are two new results: (1) If there is a simplified (ω1, 1)-morass, then there exists a ccc forcing of sizeω1that adds an ω2-Suslin tree. (2) If there is a simplified (ω1, 2)-morass, then there exists a ccc forcing of sizeω1that adds a 0-dimensional Hausdorff topologyτonω3which has spreads(τ) =ω1. While (2) is the main result of the paper, (1) is only an improvement of a previous result, which is based on a simple observation. Both forcings preserveGCH. To show that the method can be changed to produce models where CH fails, we give an alternative construction of Koszmider's model in which there is a chain 〈Xα∣α<ω2〉 such thatXα⊆ω1.Xβ–Xαis finite andXα–Xβhas sizeω1for allβ<α<ω2.

Some results on higher Suslin trees

1990 ◽  
Vol 55 (2) ◽  
pp. 526-536 ◽  
Author(s):  
R. David
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Suslin Tree

Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using ω1 Suslin trees can be extended to κ+ Suslin trees for any regular cardinal κ. Some of these are given in §1.In many such constructions, sequences of Suslin trees are used. In §II we show, in various ways, that the generalization to sequences, even ω-sequences, of κ+ Suslin trees cannot be done.In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in §III.Our notation is standard. If T is a tree and x ∈ T, then ∣x∣ is the height of x in T. We define Tα (or T(α)) = {x ∈ T: ∣x∣ = α} and T∣α = {x ∈ T: ∣x∣ < α}.If p and q are forcing conditions, p ≤ q means that p has more information than q.If (Tα: α ∈ I) is a sequence of trees, Π Tα will always mean the set of (xα: α ∈ I) such that xα ∈ Tα and ∣xα∣ = ∣xβ∣ for α, β ∈ I.For functions b, T,… we denote by b ∣α, T∣ α,… their restriction to α.If x is a sequence of ordinals and α is an ordinal, x∧α is the sequence obtained by concatenating α at the end of x.

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