The main aim of this paper is to give an example of a non-compact, hereditarily separable, locally compact, perfectly normal, countably compact space; the construction is based on Jensen's Combinatorial Principle <>. This principle is known to follow from Godel's Axiom of Constructibility and so our result is in fact a consistency theorem. It is not known whether such an example can be constructed on the assumption of the continuum hypothesis CH alone (note that 0 implies CH). Juhasz, Kunen and Rudin [14] obtained a very beautiful modification of the technique presented here which allows the construction to be derived from CH alone but with loss of countable compactness. This is most perplexing since the situation in our construction is quite the reverse: countable compactness is derived from CH and the remaining properties from a hypothesis which is apparently weaker than <>• It has since been shown, however, by W. Weiss [30] that if CH be negated and Martin's Axiom assumed, every perfectly normal, countably compact space is compact.A more complicated example of a non-compact, perfectly normal, countably compact space was independently and a little earlier given by Fedorcuk in [4] (see also [5, 6]) to illuminate certain problems in dimension theory. Further applications of the 0 technique and the CH-counterpart were studied by Rudin and Zenor [22], Fedorcuk [7, 8], J. Vaughan [28], E. Pol [19], M. Wage [29] and also by the present author in [17] and [18]. Other related matter may be found in papers of Rudin [21], van Douwen [26] and van Douwen and Wicke [27]. We wish also to call to attention R. Pol's most elegant construction in [20], which comes very close to the results of this paper without assuming any additional set-theoretic axioms. The question whether perfectly normal, countably compact spaces are compact was raised by a number of authors in various contexts e.g.: Malyhin and Shapirovskii in a draft of [16], Stephenson [24], Hager, Reynolds and Rice [10], who weakened the hypotheses of the question.In §2 we prove some theorems about countably compact spaces whose open sets are !F a and deduce a strengthening of Stephenson's main theorem in [24]. The theorems allow us to predict many properties of our example.It is a great pleasure to thank Professor M. E. Rudin for bringing the problem treated here to my attention and to thank Professor Roy O. Davies, for many, very interesting and stimulating discussions. The author gratefully acknowledges the financial support of the U.K. Science Research Council and the kind hospitality of Leicester University during the tenure of an S.R.C. Fellowship.Notation. We follow set-theoretic terminology as in, e.g., [1]; in particular, every ordinal is identified with the set of preceding ordinals. Topological terminology is as in [3] or [13]; in particular, x is a point of condensation of a set S if every neighbourhood of x meets S in a set of cardinality \S\. Other definitions are recalled in §2.

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We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski's theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

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We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation, completing our programme of reducing the number of hard proofs there to zero.S is continuous at 0 iff S(z n ) → 0, for some sequence z n ↑ 0, and then S is continuous everywhere, if finite-valued.The last part above draws on [HilP, Th. 2.5.2] that, for a subadditive function, continuity at the origin implies continuity everywhere. Theorem 0 above, in the presence of right-sided continuity, asserts that the merest hint of left-sided continuity gives full continuity; contrast this with the behaviour of the subadditive function 1 [0,∞) , which is continuous on the right but not on the left. This leads to the question of whether right-sided continuity can be thinned out. We are able to do so in the next two results below, but at the cost of imposing more structure, either on the left, or on the right. We need the following two definitions.Definitions. 1. Say that Σ is locally Steinhaus-Weil (SW), or has the SW property locally, if for x, y ∈ Σ and, for all δ > 0 sufficiently small, the sets Σ δ z := Σ ∩ B δ (z),

We give a new theory of Beurling regular variation (Part II). This includes the previously known theory of Beurling slow variation (Part I) to which we contribute by extending Bloom's theorem. Beurling slow variation arose in the classical theory of Karamata slow and regular variation. We show that the Beurling theory includes the Karamata theory.

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To Ranko Bojani´c on his 90 th birthday.Abstract. The Cauchy functional equation is not only the most important single functional equation, it is also central to regular variation. Classical Karamata regular variation involves a functional equation and inequality due to Goldie; we study this, and its counterpart in Beurling regular variation, together with the related Go÷¾ ab-Schinzel equation.