Kohzo Yamada

Frechet-Urysohn spaces in free topological groups

Let F(X) and A(X) be respectively the free topological group and the free Abelian topological group on a Tychonoff space X. For every natural number n we denote by F n (X) (A n (X)) the subset of F(X) (A(X)) consisting of all words of reduced length 3 except for n = 4. In addition, however, there is a first countable, and hence Frechet-Urysohn subspace Y of F(X) (A(X)) which is not contained in any F n (X) (A n (X)). We shall show that if such a space Y is first countable, then it has a special form in F(X) (A(X)). On the other hand, we give an example showing that if the space Y is Frechet-Urysohn, then it need not have the form.

Prime subspaces in free topological groups

Abstract Let F(X) (A(X)) be the free (Abelian) topological group over X. We prove: If P is one of the spaces R , Q , R , Q , βω, βω ω and 2/gk for an infinite κ and if F(X) or A(X) contains a copy of P, then X contains a copy of P. If P is the one-point compactification of an infinite discrete space or ω1 + 1, this is not true. If P = ω1, this holds for F(X) but is independent of ZFCfor A(X).

Free topological groups on metrizable spaces and inductive limits

Abstract We prove that for a metrizable space X the following are equivalent: (i) the free Abelian topological group A(X) is the inductive limit of the sequence {A n (X):n∈ N } , where A n (X) is formed by all words of reduced length ≤n ; (ii) X is locally compact and the set of all non-isolated points of X is separable. In the non-Abelian case, for a metrizable X the following are equivalent: (i) the free topological group F(X) is the inductive limit of the sequence {F n (X):n∈ N } ; (ii) X is either locally compact separable or discrete.

Subsets of Rn with convex midsets

Abstract Let R n be the Euclidean space with the standard metric d . In this paper, we prove the following: 1. Let n ⩾3 and X be a nondegenerate, i.e., containing more than one point, compact subset of R n such that for each pair x,y of distinct points of X , the midset M ( x , y )={ z ∈ X : d ( x , z )= d ( y , z )} is a boundary of a convex ( n −1)-cell. Then X is a boundary of a convex n -cell. 2. Let n ⩾2 and X be a nondegenerate subset of R n such that for each pair x,y of distinct points of X , M ( x,y ) is a convex ( n −1)-cell. Then X is a convex n -cell.