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

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1. CMB 2013 (vol 57 pp. 335)

Karassev, A.; Todorov, V.; Valov, V.
Alexandroff Manifolds and Homogeneous Continua
ny homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.

Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuum
Categories:54F45, 54F15

2. CMB 2011 (vol 55 pp. 378)

Oman, Greg; Salminen, Adam
On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality
Let $R$ be a commutative ring with identity, and let $M$ be a unitary module over $R$. We call $M$ H-smaller (HS for short) if and only if $M$ is infinite and $|M/N|<|M|$ for every nonzero submodule $N$ of $M$. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: suppose $M$ is faithful over $R$, $R$ is a domain (we will show that we can restrict to this case without loss of generality), and $K$ is the quotient field of $R$. If $M$ is HS over $R$, then $R$ is HS as a module over itself, $R\subseteq M\subseteq K$, and there exists a generating set $S$ for $M$ over $R$ with $|S|<|R|$. We use this result to generalize a problem posed by Kaplansky and conclude the paper by answering an open question on Jónsson modules.

Keywords:Noetherian ring, residually finite ring, cardinal number, continuum hypothesis, valuation ring, Jónsson module
Categories:13A99, 13C05, 13E05, 03E50

3. CMB 2011 (vol 54 pp. 244)

Daniel, D. ; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
Homogeneous Suslinian Continua
A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum $X$ has the property that the set of points at which $X$ is connected im kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.

Keywords:connected im kleinen, homogeneity, Suslinian, locally connected continuum
Categories:54F15, 54C05, 54F05, 54F50

4. CMB 2005 (vol 48 pp. 195)

Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
On Suslinian Continua
A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most $\omega_1$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finite
Categories:54F15, 54D15, 54F50

5. CMB 1999 (vol 42 pp. 13)

Brendle, Jörg
Dow's Principle and $Q$-Sets
A $Q$-set is a set of reals every subset of which is a relative $G_\delta$. We investigate the combinatorics of $Q$-sets and discuss a question of Miller and Zhou on the size $\qq$ of the smallest set of reals which is not a $Q$-set. We show in particular that various natural lower bounds for $\qq$ are consistently strictly smaller than $\qq$.

Keywords:$Q$-set, cardinal invariants of the continuum, pseudointersection number, $\MA$($\sigma$-centered), Dow's principle, almost disjoint family, almost disjointness principle, iterated forcing
Categories:03E05, 03E35, 54A35

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