This is a brief survey of the use of transfinite induction in metric fixed-point theory. Among the results discussed in some detail is the author’s 1989 result on directionally nonexpansive mappings (which is somewhat sharpened), a result of Kulesza and Lim giving conditions when countable compactness implies compactness, a recent inwardness result for contractions due to Lim, and a recent exte...

For continuous time Markov chains on a countable state space, we derive a parabolic Harnack inequality using probabilistic methods. We derive some consequences of this inequality for the compactness of parabolic (i.e. spacetime harmonic) functions of the process.

Abstract. A new form of α-compactness is introduced in L-topological spaces by α-open L-sets and their inequality where L is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice L. It can also be characterized by means of α-closed L-sets and their inequality. When L is a completely distributive de Morgan algebra, its many characterizations are presented and the re...

Arhangel’skĭı [3] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper. The concept of countable core in [3] is a little hard to understand at first; Arhangel’skĭı, however, provides equivalents which are easier to understand, and so we will take one of...

• The conditions are investigated under which the upper Vietoris topology coincides with Scott on K ( X ) . For a well-filtered space having first-countable Smyth power space, and coincide. T 0 in set of minimal elements any compact saturated subset, its is first-countable. Alexandroff double circle, first-countable, not , let be poset all non-empty sets reverse inclusion order. said to have pr...