The purpose of this paper is to introduce the concepts of semi*-connected spaces, semi*-compact spaces and semi*-Lindelof spaces. We investigate their basic properties. We also discuss their relationship with already existing concepts. Mathematics Subject Classification: 54D05, 54D30.

0.1. This paper presents an investigation of the following problem. Exhibit a class X of topological spaces which contains all peano spaces and which has the following properties: (1) a cyclic element theory exists in each space of the class, (2) the abstract set consisting of all cyclic element of any space X of the class can be topologized so as to be a member of the class X, and (3) the hype...

We construct, assuming Jensen’s principle ♦, a one-dimensional locally connected hereditarily separable continuum without convergent sequences.

A finite distance space X , d d : X 2 --,7/ is hypermetric (of negative type) if axay d(x, y) < 0 for all integral sequences {axlx ~ X} that sum to 1 (sum to 0). X, d is connected if the set {(x, y)ld(x, y) = 1, x, y ~ X} is the edge set for a connected graph on X, and graphical if d is the path length distance for this graph. Then we prove Theorem 1. A connected space X, d has negative type if...

Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence...

The non-Hausdorff suspension of the one-sphere S1 of complex numbers fails to model the group’s continuous multiplication. Moreover, finite connected H-spaces are contractible, and therefore cannot model infinite connected non-contractible H-spaces. For an H-space and a finite model of the topology, the multiplication can be realized on the finite model after barycentric subdivision.

We construct, assuming Jensen’s principle ♦, a one-dimensional locally connected hereditarily separable continuum without convergent sequences.

We first describe Krull-Schmidt theorems decomposing H spaces and simply-connected co-H spaces into atomic factors in the category of pointed nilpotent p-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic H spaces and homotopy types of atomic co-H spaces, and construct a split fibration which connects them and illuminates the decomposi...