In nonstandard topology we can describe a given topology by the neighbourhood-monads of the standard points (see Sari, chapter 6 in Diener and Diener [2]). Calling an internal point near-standard if it is contained in the neighbourhood-monad of some standard point and remote otherwise we give some statements about locally finite families and paracompact spaces in terms of nearstandard and remot...

We introduce topological definitions of expansivity, shadowing, and chain recurrence for homeomorphisms. They generalize the usual definitions for metric spaces. We prove various theorems about topologically Anosov homeomorphisms (maps that are expansive and have the shadowing property) on noncompact and non-metrizable spaces that generalize theorems for such homeomorphisms on compact metric sp...

Let (X, z) be a topological space, and (Y, d) a metric space. A mapping F: X -+ 2’ which associates with every x E X a non-empty subset F(x) OF Y is said to be lower semi-continuous (I.s.c.) (respectively, upper semi-continuous (u.s.c.)) if, for each open set % in Y, the set (X E X: F(x) n %f # @ > (respectively, the set (X E X: F(x) c @ )) is open in X A mapping f: X-a Y is a selection for F i...

We study spaces that are continuous images of the usual space [0, ω1) of countable ordinals. We begin by showing that if Y is such a space and is T3 then Y has a monotonically normal compactification, and is monotonically normal, locally compact and scattered. Examples show that regularity is needed in these results. We investigate when a regular continuous image of the countable ordinals must ...

We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomor-phic to the measure algebra of 2 ! 1. The rst construction uses } to produce an S-space with no convergent sequences in which every perfect set is a G. A space with these properties must be both hereditarily normal and hereditarily countably paracompact. The sec...

We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC. 0. Introduction. This paper has been motivated by several unsolved problems in general topology and topological algebra. The co...

In recent years, some generalized structures of topology were introduced. Supra was one the most important those generalizations. To contribute in this orientation, we devoted work to studying limit points and separation axioms on supra topological spaces by using $b$-open sets. We define them a similar way their counterparts spaces. general, demonstrate main properties investigate sufficient c...