This paper presents a novel approach to the reconstruction of geometric models and surfaces from given sets of points using volume splines. It results in the representation of a solid by the inequality The volume spline is based on use of the Green's function for interpolation of scalar function values of a chosen "carrier" solid. Our algorithm is capable of generating highly concave and branching objects automatically. The particular case where the surface is reconstructed from cross-sections is discussed too. Potential applications of this algorithm are in tomography, image processing, animation and CAD f o r bodies with complex surfaces.

A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X) ℵ 0 , where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P-space, then the character of the free abelian topological group is bounded below by the "small cardinal" d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.

We consider special subclasses of the class of Lindelöf -spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class L ( κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ ω. In the case κ = ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P -approximable spaces considered by Tkačenko. The case κ = 1 corresponds to the spaces of countable network weight, but even the case κ = 2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ 1 is in the class L ( ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in L ( ω) have dense metrizable subspaces, partially answering a question of Tkačenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions. PreliminariesAll spaces we consider are Tychonoff (that is, completely regular Hausdorff), unless otherwise indicated. We use terminology and notation as in [5], with the exception that the tightness of a space X is denoted as t (X).Given a locally compact space X, we denote by αX its one-point compactification, the new point will be usually denoted by ∞ (unless ∞ ∈ X). The one-point compactification of a discrete space of size κ will be denoted by A(κ).

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