Local compactness in free topological groups

Peter Nickolas, Mikhail Tkachenko
2003 Bulletin of the Australian Mathematical Society  
We show that the subspace A n (X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n € u> if and only if A2(X) is locally compact if and only if ^( X ) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace F n (X) of the free topological group F(X) is locally compact for each n G w if and only if Fn(X) is locally compact if and only if F n (X) has pointwise countable
more » ... ntwise countable type for each n e w i f and only if F^{X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that A n {X) has pointwise countable type for each n € w if and only if A2(X) has pointwise countable type if and only if •FM-Y) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that Fi(X) is locally compact if and only if F3(X) is locally compact, and that Fz(X) has pointwise countable type if and only if F$(X) has pointwise countable type. subject to the Cambridge Core terms of use, 244 P. Nickolas and M. Tkachenko [2] For every non-negative integer n, let F n (X) be the subset of F(X) which consists of all elements having reduced length less than or equal to n with respect to the basis X, and define the subset A n (X) of A(X) in a similar way. Yamada proved in [22, Theorem 4.9] that F n (X) is metrisable for each integer n € u i f and only if F 4 (X) is metrisable if and only if X is compact metrisable or discrete. The Abelian case is different: A n (X) is metrisable for each integer n if and only if A 2 (X) is metrisable if and only if X is metrisable and the set X' of non-isolated points of X is compact. We conclude in particular that the metrisability of all finite levels F n (X) of the group F(X) does not imply the metrisability of F(X), and that neither does the metrisability of the levels A n (X) imply that of A{X). (One can, however, see this easily by a direct argument, since results known to Graev [6] show that if X is an infinite compact metric space, then the finite levels of F(X) and A(X) are compact and metrisable, while the groups themselves are not metrisable.) The local compactness of the spaces F n (X) was considered by Yamada under the additional assumption that X is metrisable. He showed in [21, Proposition 3.3] that, under this assumption, the levels F n (X) are locally compact if and only if X is either compact or discrete. Afterwards, Pestov and Yamada proved in [15] that, for an arbitrary TychonofT space X, if each F n (X) is locally compact, then X is discrete or pseudocompact. In fact, pseudocompactness in this assertion can be strengthened to countable compactness [15, Coroollary 3.6]. Here we show that the spaces F n (X) are locally compact for all n G w if and only if F^X) is locally compact if and only if X is either discrete or compact, thus finishing the work started in [21, 15] . Exactly the same conclusion remains valid if one weakens local compactness to pointwise countable type (see Theorem 2.9). It turns out that n = 4 is the minimal integer with this property, in both cases: by Theorem 2.13 and Proposition 2.14, F 2 {X) is locally compact if and only if F 3 (X) is locally compact if and only if X is the topological sum of a compact space and a discrete space, and F 2 (X) has pointwise countable type if and only if F 3 (X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the points of the complement X \C are isolated in X (see Theorem 3.3 and Proposition 3.7). Further, F 2 (X) is locally compact if and only if A 2 (X) is locally compact if and only if A n (X) is locally compact for all n € u>, with a similar equivalence holding in the case of pointwise countable type (Theorems 2.13 and 3.5). Finally, it turns out that F 2 (X) has pointwise countable type if and only if F 2 (X) is Cech-complete, and the same equivalence holds in the Abelian case (Theorem 3.8). This article is organised as follows. Section 1.1 introduces notation and terminology used throughout the paper. In Section 1.2 we present in summary form a few basic results about free (Abelian) topological groups which have several applications in the main body of the article. The local compactness and pointwise countable type of the finite levels of the groups F(X) and A(X) are studied in Section 2. We show in Lemma 2.1 that if available at https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0004972700037631 fatcat:4zitfq4fo5ddfmbn6q3liqz6ry