Yakovlev [On bicompacta in -products and related spaces, Comment. Math. Univ. Carolin. 21.2 (1980), 263–283] showed that any Eberlein compactum is hereditarily -metacompact. We show that this property actually characterizes Eberlein compacta among compact spaces of finite metrizability number. Uniformly Eberlein compacta and Corson compacta of finite metrizability number can be characterized in an analogous way.
We prove that every planar rational compactum with rim-type ≤ α, where α is a countable ordinal greater than 0, can be topologically embedded into a planar rational (locally connected) continuum with rim-type ≤ α.
Let X be a connected locally compact metric space. It is known that if X is locally connected, then the space of ends (Freudenthal ends), EX, can be represented as the inverse limit of the set (space) S(X C) of components of X C, i.e., as the inverse limit of the inverse system . In this paper, the above result is significantly improved. It is shown that for a space which is not locally connected, we can replace the space of components by the space of quasicomponents Q(X C) of X C. The following...
For mappings , where is a merotopic space equipped with a diameter function, we introduce and examine an entropy, called the -entropy. The topological entropy and the entropy of self-mappings of metric spaces are shown to be special cases of the -entropy. Some connections with other characteristics of self-mappings are considered. We also introduce and examine an entropy for subsets of , which is closely connected with the -entropy of .
We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.
We present a result which affords the existence of equivalent metrics on a space having distances between certain pairs of points predetermined, with some restrictions. This result is then applied to obtain metric spaces which have interesting properties pertaining to the span, semispan, and symmetric span of metric continua. In particular, we show that no two of these variants of span agree for all simple closed curves or for all simple triods.