MacLane’s planarity criterion states that a finite graph is planar if and only if its cycle space has a basis B such that every edge is contained in at most two members of B. Solving a problem of Wagner (1970), we show that the topological cycle space introduced recently by Diestel and Kühn allows a verbatim generalisation of MacLane’s criterion to locally finite graphs. This then enables us to...

The necessity of a theory of General Topology and, most of all, of Algebraic Topology on locally finite metric spaces comes from many areas of research in both Applied and Pure Mathematics: Molecular Biology, Mathematical Chemistry, Computer Science, Topological Graph Theory and Metric Geometry. In this paper we propose the basic notions of such a theory and some applications: we replace the cl...

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodesic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodesic cycles need not generate C(G).

We show that any locally finite abelian variety is generated by a finite algebra. We solve a problem posed by D. Hobby and R. McKenzie by exhibiting a nonfinitely based finite abelian algebra.

Let G be an uncountable universal locally finite group. We study subgroups H < G such that for every g ∈ G, |H : H ∩H| < |H|.

DEFINITION. A group G is called a universal locally finite central extension of A provided that the following conditions are satisfied. (i) A <= (G (the centre of G). (ii) G is locally finite. (iii) (/1-injectivity). Suppose that A <= B <= D with A a (D, that D/A is finite, and that q>: B -> G is an ^-isomorphism (that is, q>{a) = a for all as A). Then there exists an extension q>: D -*• G of (...