Metrizability of Clifford topological semigroups

We prove that a topological Clifford semigroup $S$ is metrizable if and only if $S$ is an $M$-space and the set $E=\{e\in S:ee=e\}$ of idempotents of $S$ is a metrizable $G_\delta$-set in $S$. The same metrization criterion holds also for any countably compact Clifford topological semigroup $S$.Comment: 4 page

Competing with stationary prediction strategies

In this paper we introduce the class of stationary prediction strategies and construct a prediction algorithm that asymptotically performs as well as the best continuous stationary strategy. We make mild compactness assumptions but no stochastic assumptions about the environment. In particular, no assumption of stationarity is made about the environment, and the stationarity of the considered strategies only means that they do not depend explicitly on time; we argue that it is natural to consider only stationary strategies even for highly non-stationary environments.Comment: 20 page

The Collins-Roscoe mechanism and D-spaces

We prove that if a space X is well ordered $(\alpha A)$, or linearly semi-stratifiable, or elastic then X is a D-space

The well-ordered (F) spaces are D-spaces

We studied the relationships between Collins-Roscoe mechanism and D-spaces, proved that well-ordered (F) spaces are D-spaces. This positively answered a question asked by D.Soukup and Y.Xu before.Comment: 6 page

Heavy metals in the irrigation water, soils and vegetables in the Philippi horticultural area in the Western Cape Province of South Africa

The aims of this study were to investigate the extent of heavy metal contamination in the Philippi horticultural area in the Western Cape Province, South Africa. Concentrations of Cd, Cr, Cu, Mn, Ni, Pb and Zn were determined in the irrigation water, soils and vegetables in both winter and summer cropping seasons with an ICP-AES and tested against certified standards. Differences were found in heavy metal concentrations between the winter and summer cropping seasons in the irrigation water, soils and vegetables. Certain heavy metals exceeded the maximum permissible concentrations in the irrigation water, soils and vegetables produced in South Africa. These toxic concentrations were predominantly found in the summer cropping season for the soils and in the crops produced in winter. It is thus suggested that further studies are carried out in the Philippi horticultural area to determine the sources of the heavy metals to try and mitigate the inputs thereof and therefore reduce the amount of heavy metals entering the human food chain.ISI & Scopu

Thin-Type Dense Sets and Related Properties

We build on Gruenhage, Natkaniec, and Piotrowskiʼs study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that Formula Not Shown has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin–Keisler order on βω. We also introduce and study the concepts of \u3cκ-thin and superslim dense sets

Under consideration for publication in Math. Struct. in Comp. Science Quotients of countably based spaces are not

Dedicated to Klaus Keimel on the occasion of his 65th birthday In this note we show that quotients of countably based spaces (qcb spaces) and topological predomains as introduced by M. Schröder and A. Simpson are not closed under sobrification. As a consequence replete topological predomains need not be sober, i.e. in general repletion is not given by sobrification. Our counterexample also shows that a certain tentative “equalizer construction ” of repletion fails for qcb spaces. Our results extend also to the more general class of core compactly generated spaces