Roitman's combinatorial principle $\Delta$ is equivalent to monotone normality of the nabla product, $\nabla (\omega +1)^\omega$. If $\{ X_n : n\in \omega\}$ a family metrizable spaces and $\nabla_n X_n$ monotonically normal, then hereditarily paracompact. Hence, if holds box product $\square +1)^\omega$ Large fragments hold in $\mathsf{ZFC}$, yielding large subspaces (\omega+1)^\omega$ that ar...

Xizhong Zheng∗1,2, Robert Rettinger3, and George Barmpalias4 1 Theoretische Informatik, BTU Cottbus, 03044 Cottbus, Germany. [email protected] 2 Dept. of Comput. Sci., Jiangsu University, Zhenjiang 212013, China 3 Theoretische Informatik II, FernUniversität Hagen, 58084 Hagen, Germany. [email protected] 4 School of Mathematics University of Leeds, Leeds LS2 9JT, U.K...

Monotonically metacompact spaces were recently introduced as an extension of the concept of monotonically compact spaces. In this note we answer a question of Popvassilev, and Bennett, Hart, and Lutzer, by showing that every compact, Hausdorff, monotonically (countably) metacompact space is metrizable. We also show that certain countable spaces fail to be monotonically (countably) metacompact.

In this paper, we consider the existence of extremal solutions for nonlinear fourth-order differential equation. By use a new comparison result, some sufficient conditions are established by combining monotone iterative technique and methods lower upper solutions. Finally, an example is given to illustrate validity our main results.

Article history: Received 17 July 2008 Received in revised form 6 October 2009 MSC: primary 54D20 secondary 54E30, 54E35, 54F05