In [4-8], the author has obtained several counterexamples to conjectures involving chain conditions, normality conditions, completeness, and the existence of point countable bases in Moore spaces. Each of these examples was obtained by constructing, by various means, a Moore space based on another space X0. In this paper, the author unifies these construction techniques and states some of the relationships between the original spaces and the derived Moore spaces.

Let X and Y denote polyhedra, i : X → Y a PL immersion. A regular neighborhood of X associated with i is a regular neighborhood (e, Ri(X)) of X together with an immersion j : Ri(X) → Y

such that the diagram.

X is a set and End X is a semigroup, under composition, of functions, which map X into X. We characterize those elements of End X which are regular and then we completely determine Green's relations for these elements. The conditions we place on End X are sufficiently mild to permit such semigroups as S(X), the semigroup of all continuous self maps of a topological space X and L(V), the semigroup of all linear transformations on a vector space V, to be regarded as special cases.

How large a matching must a graph have?

We consider graphs G (finite, undirected, with no loops or multiple edges), with order nG (always ≧ 1) and mG the maximum number of edges in a matching of G. The matchability μG of G is the fraction (2m/n) of nodes covered by a maximum matching.

All groups in this paper are assumed to be finite.

Let G be a group with op(G) ≠ 1 which is p-constrained and p-stable, p odd. If P is an Sp-subgroup of G, then by Glaubermaris Theorem, [3, 8.2.11],

G = Op′(G)NG(ZJ(P)).

An abelian topological group can be considered simply as an abelian group or as a topological space. The question considered in this article is whether the topological group structure is determined by these weaker structures. Denote homeomorphism, isomorphism, and homeomorphic isomorphism by ≈, ≅ , and =, respectively. The principal results are these.

Theorem 1. If G1andG2are locally compact and connected, then G1≈ G2implies G1= G2.

The translational hull is of central importance in the construction of ideal extensions and the study of densely embedded ideals particularly for weakly reductive semigroups (see [4, Chapter III]). The translational hull of semigroups belonging to a few special classes is known in an explicit form, and for some other classes of semigroups, certain properties of their translational hulls have been established (see [4, Chapter V]). We have generalized in [5] the concept of an inverse limit of groups in order to give a construction of the translational hull of a semigroup which is a semilattice of groups.