Abstract
Let X be a metrizable space and F( X) and A( X) be the free topological group over X and the free Abelian topological group over X respectively. We establish the following criteria: 1. (a) tightness of A( X) is countable if the set X′ of all nonisolated points in X is separable, 2. (b) tightness of F( X) is countable if is separable or discrete, 3. (c) A( X) is a k-space iff X is locally compact and X′ is separable, 4. (d) F( X) is a k-space iff X is locally compact separable or discrete. We also show that if X is second-countable, then F( X) and A( X) are k R -spaces iff X is locally compact.
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