### Citations

106 |
Set theory: on the structure of the real line
- Bartoszynski, Judah
- 1995
(Show Context)
Citation Context ...n [Gol93, 6.33] showed that the Laver property is preserved by countable support proper iterations. It is also known that if a forcing has the Laver property then it does not add any Cohen reals (see =-=[BJ95]-=-). In [BH04], B. Balcar and M. Hrušák show that h (R∗) ≤ min {h, add(M)}. Since forcing with MR∗ we have h (R ∗) = ℵ2 then our forcing MR∗ must add Cohen reals and therefore it does not have the Lav... |

44 |
The space of ultrafilters on covered by nowhere dense sets
- Balcar, Simon
(Show Context)
Citation Context ...ogists who studied this characteristic for general topological spaces. J. van Mill and S. Williams in [vMW83] introduced the definition of the weak Novák number. B. Balcar, J. Pelant and P. Simon in =-=[BPS80]-=- studied the Novák number of N∗ using that N∗ always has a tree pi-base. In general, a tree pi-base for a topological space is a pi-base which forms a tree when ordered by reverse inclusion. They als... |

30 | Tools for your forcing construction. In Set theory of the reals (Ramat - Goldstern - 1991 |

14 |
On cardinal invariants of the continuum. In Axiomatic set theory
- Shelah
- 1983
(Show Context)
Citation Context ...his means that a tree pi-base for N∗ in this model has barely height ω2. Dordal’s model was actually constructed using a modified support for Mathias forcing but was easily adapted once Shelah showed =-=[She84]-=- that the “not filling towers” property — the key of Dordal’s proof — is preserved by countable support proper iterations. Shelah and Spinas [SS00] showed that in the Mathias model, h(N∗ ×N∗) = ℵ1 (bu... |

13 | and improper forcing. Second edition - Shelah, Proper - 1998 |

6 |
A model in which the base-matrix tree cannot have cofinal branches
- Dordal
- 1987
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Citation Context ...ntable support iteration of Mathias forcing) P(ω)upslope fin has no dense tree of height ω1, so h(N∗) = ℵ2 in the Mathias model. However, while there is no dense tree of height ω1, P.L. Dordal showed =-=[Dor87]-=- that there must be a dense tree in which there are no branches of length ω2. This means that a tree pi-base for N∗ in this model has barely height ω2. Dordal’s model was actually constructed using a ... |

4 | The distributivity numbers of P(ω)/ fin and its square
- Shelah, Spinas
(Show Context)
Citation Context ...s forcing but was easily adapted once Shelah showed [She84] that the “not filling towers” property — the key of Dordal’s proof — is preserved by countable support proper iterations. Shelah and Spinas =-=[SS00]-=- showed that in the Mathias model, h(N∗ ×N∗) = ℵ1 (but h(N∗) = ℵ2 holds). It was known that h(R∗) ≤ h(N∗), and Alan Dow proved that in the Mathias model h(R∗) = ℵ1. Later, armed with a better understa... |

3 |
Distributivity of the algebra of regular open subsets of βR
- Balcar, Hrušák
(Show Context)
Citation Context ...he Mathias model h(R∗) = ℵ1. Later, armed with a better understanding of the cardinal h(R∗), B. Balcar and M. Hrušák proved the following result from which Dow’s result easily follows: Theorem 1.2 (=-=[BH04]-=-). h(R∗) ≤ {h, add(M)}, where M is the ideal of meager sets in R, and add(M) its additivity number. Dordal’s result implies that h(N∗) = n(N∗) = ℵ2 in the Mathias model. However it is shown in [BPS80]... |

2 |
Tree pi-bases for βN−N in various models
- Dow
- 1989
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Citation Context ...’s result implies that h(N∗) = n(N∗) = ℵ2 in the Mathias model. However it is shown in [BPS80] that n(R∗) is always at least ℵ2, hence, in the Mathias model, h(R∗) < n(R∗). Nevertheless, Theorem 1.3 (=-=[Dow89]-=-). There is a model for ZFC in which h(R∗) = n(R∗). 2. Modification of Mathias forcing Our terminology is mostly standard. The bar over a set denotes the closure of it with respect to the ambient spac... |

2 |
S.W.,A compact F -space not co-absolute with βN−N, Topology Appl
- Mill, Williams
- 1983
(Show Context)
Citation Context ... Novák number . n(X) is also known as the Baire number of X . J. Novák was among the first topologists who studied this characteristic for general topological spaces. J. van Mill and S. Williams in =-=[vMW83]-=- introduced the definition of the weak Novák number. B. Balcar, J. Pelant and P. Simon in [BPS80] studied the Novák number of N∗ using that N∗ always has a tree pi-base. In general, a tree pi-base f... |

1 |
The regular open algebra of βR\R is not equal to the completion of P(ω
- Dow
- 1998
(Show Context)
Citation Context ... 122437 and PAPIIT grant IN106705. 722 F.Hernández-Hernández dense open subsets of Xγ and their intersection has empty interior. We have been interested in the following questions which appeared in =-=[Dow98]-=-: Is h ((R∗)ω) = h (R∗)? Is it true that h(R∗) ≤ h(N∗ × N∗)? Is it true n(R∗) = n(N∗)? Alan Dow has conjectured positive answers for the first two questions; the third one is due to E.K. van Douwen. F... |

1 |
The distributivity numbers of finite products of P(ω
- Shelah, Spinas
- 1998
(Show Context)
Citation Context ...ether it is possible to have h(R∗ × R∗) < h(R∗). By the result in the last section MR∗ seemed hopeful to use it with the methods of Shelah and Spinas [SS00]. Some time later we read their second part =-=[SS98]-=- where they claim that the important properties of Mathias forcing which are essential to their proof are: (1) Mathias forcing factors into a ℵ1-closed and a σ-centred forcing; (2) Mathias forcing is ... |

1 |
spaces, and coabsolutes of βN
- Williams, Gleason
- 1982
(Show Context)
Citation Context ...base. In general, h(X) is the distributivity number of the Boolean algebra RO(X); equivalently, it is the minimum cardinal κ such that forcing with RO(X) does not add a new subset of κ. By results in =-=[Wil82]-=-, for every locally compact noncompact metric space X , its Stone-Čech remainder, X∗, always has a tree pi-base, h(X∗) = wn(X∗) and this cardinal coincides with the minimum height of a tree pi-base. ... |