countable chain condition
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2021 ◽  
Vol 13 (1) ◽  
pp. 81-88
Author(s):  
I. Krasikova ◽  
M. Pliev ◽  
M. Popov

We study measurable elements of a Riesz space $E$, i.e. elements $e \in E \setminus \{0\}$ for which the Boolean algebra $\mathfrak{F}_e$ of fragments of $e$ is measurable. In particular, we prove that the set $E_{\rm meas}$ of all measurable elements of a Riesz space $E$ with the principal projection property together with zero is a $\sigma$-ideal of $E$. Another result asserts that, for a Riesz space $E$ with the principal projection property the following assertions are equivalent. (1) The Boolean algebra $\mathcal{U}$ of bands of $E$ is measurable. (2) $E_{\rm meas} = E$ and $E$ satisfies the countable chain condition. (3) $E$ can be embedded as an order dense subspace of $L_0(\mu)$ for some probability measure $\mu$.


2014 ◽  
Vol 90 (3) ◽  
pp. 521-524 ◽  
Author(s):  
WEI-FENG XUAN ◽  
WEI-XUE SHI

AbstractWe prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ is a space satisfying the discrete countable chain condition with a rank 3-diagonal then the cardinality of $X$ is at most $\mathfrak{c}$.


2014 ◽  
Vol 90 (1) ◽  
pp. 141-143 ◽  
Author(s):  
WEI-FENG XUAN ◽  
WEI-XUE SHI

AbstractWe prove that if a space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ with a rank 2-diagonal either has the countable chain condition or is star countable then the cardinality of $X$ is at most $\mathfrak{c}$.


Author(s):  
V. V. Mykhaylyuk

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.


2006 ◽  
Vol 144 (1-3) ◽  
pp. 107-116 ◽  
Author(s):  
Lúcia R. Junqueira ◽  
Paul Larson ◽  
Franklin D. Tall

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