On $LC^n$ metric spaces

Test Spaces for Normal Spaces

“…The case « = 1 requires separate treatment because of the non-abelian character of 7Ti(F). By hypothesis, F is a compact 1dimensional, 1-LC metric space, so by Lemma 3 and Kodama's result [5] Fis an ANR(metric).…”

“…A space Y is called an absolute neighborhood retract relative to the class Q (abbreviated ANR(Ç)) if (a) Y is in Q, and (b) whenever Y is imbedded topologically as a closed subset of a Q-space z, then F is a retract of some neighborhood of Y in Z (see [3]). Kodama has shown in [5] that an «-dimensional metric space F is an ANR (metric) if and only if Y is LC".…”

On Closed Images of the Space of Irrationals

“…In fact, we actually prove a more general result which does not assume separability. The proof is based on a refinement of a known lemma (see [5, p. 281 ], and [2] or [4] for the nonseparable case) that every nonempty closed subset of a metric space X of covering dimension 0 is a retract of X. The refinement, which seems to be new even in the separable case, asserts the existence of a closed retraction.…”

Test Spaces for Metric Spaces

“…Insofar as metric spaces are concerned, this definition is equivalent to the one employed by Kodama in [9], so that his results are applicable here. In this paper, dim X will always mean the dimension of X defined in terms of finite open coverings.…”