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Search: All articles in the CJM digital archive with keyword continuum

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1. CJM 2011 (vol 63 pp. 1416)

Shelah, Saharon
MAD Saturated Families and SANE Player
We throw some light on the question: is there a MAD family (a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any two is finite) that is saturated (completely separable \emph{i.e.,} any $X \subseteq \mathbb{N}$ is included in a finite union of members of the family \emph{or} includes a member (and even continuum many members) of the family). We prove that it is hard to prove the consistency of the negation: (i) if $2^{\aleph_0} \lt \aleph_\omega$, then there is such a family; (ii) if there is no such family, then some situation related to pcf holds whose consistency is large (and if ${\mathfrak a}_* \gt \aleph_1$ even unknown); (iii) if, \emph{e.g.,} there is no inner model with measurables, \emph{then} there is such a family.

Keywords:set theory, MAD families, pcf, the continuum
Categories:03E05, 03E04, 03E17

2. CJM 2009 (vol 62 pp. 182)

Prajs, Janusz R.
Mutually Aposyndetic Decomposition of Homogeneous Continua
A new decomposition, the \emph{mutually aposyndetic decomposition} of homogeneous continua into closed, homogeneous sets is introduced. This decomposition is respected by homeomorphisms and topologically unique. Its quotient is a mutually aposyndetic homogeneous continuum, and in all known examples, as well as in some general cases, the members of the decomposition are semi-indecomposable continua. As applications, we show that hereditarily decomposable homogeneous continua and path connected homogeneous continua are mutually aposyndetic. A class of new examples of homogeneous continua is defined. The mutually aposyndetic decomposition of each of these continua is non-trivial and different from Jones' aposyndetic decomposition.

Keywords:ample, aposyndetic, continuum, decomposition, filament, homogeneous
Categories:54F15, 54B15

3. CJM 2009 (vol 61 pp. 604)

Hart, Joan E.; Kunen, Kenneth
First Countable Continua and Proper Forcing
Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight $\aleph_1$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

Keywords:connected space, Continuum Hypothesis, proper forcing, irreducible map
Categories:54D05, 03E35

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