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IntroductionThe aim of this paper is a clarification and extension of the work of V. I. Ponomarev ([8]) on projective spectra. Ponomarev characterizes paracompact Hausdorff spaces as the spectral limits of inverse systems of simplicial complexes. In this paper the spectral limit of an inverse system of simplicial complexes is replaced by the true limit in the categorical sense of an inverse system of certain quasi-order spaces called special quasi-order spaces. In this way a characterization of the wider class of paracompact regular spaces is obtained (Theorem 3.2). In addition there is a characterization of normal regular spaces and the strongly paracompact regular spaces are identified amongst the paracompact spaces. The main technique in the proof of the theorems is the association of two 'dual' quasi-orders with each covering of a space. These are studied in § 2. Special quasi-order spaces have a very simple structure particularly from the point of view of dimension theory and the author will exploit this in subsequent work on the dimension of paracompact spaces. The definition of a special quasi-order space is given in § 2. Some properties of normal, paracompact and strongly paracompact spaces are described in § 1; these are easily obtained from well-known results ([7], [8], [9]). In the final paragraph the relation of this work to Ponomarev's theory of spectra is examined.The author wishes to thank the referee for helpful comments on an earlier version of this paper. Canonical coverings, normality and paracompactnessA canonical covering of a topological space X is a locally finite closed covering {F A } AeA of X such that for each X e A, F x = G x where {G x } is a collection of mutually disjoint open sets. Finite canonical coverings were introduced by P. S. Alexandrav and V. I. Ponomarev ([3]). Canonical coverings have been considered subsequently by Ponomarev ([8]) and others. For lack of a suitable reference we derive in the following three lemmas those properties of canonical coverings we shall need later.

A topological space is said to be finitistic if each open cover has a finitedimensional open refinement. Thus each compact space is finitistic, as is each finitedimensional paracompact space [6]. Also, since a finite-dimensional covering is point-finite, each finitistic space is weakly paracompact [7, Definition 2.2.6].The concept of finitisticity was introduced by Swan [8] for work in cohomological dimension theory. Bredon [1] found that finitistic spaces provided an appropriate setting for the study of compact Lie group actions. Recently Deo and Tripathi showed [2] that if X is a paracompact Hausdorff finitistic space and G is a compact Lie group acting on X then the orbit space X/G is also finitistic.Clearly there are finitistic spaces which are not finite-dimensional-any compact infinite-dimensional space provides an example. Let us say that a topological space is completely finitistic if every subspace is finitistic. For example, every finitedimensional hereditarily paracompact Hausdorff space is completely finitistic, since each subspace is paracompact and finite-dimensional [3,4]. The purpose of this paper is to show that every completely finitistic normal Hausdorff space is finitedimensional. LEMMA. IfW = [j V p where (V p ) pef4 is a disjoint collection of open sets of W andpe N dim V p > n p for each p, where (n p ) peN is a strictly increasing sequence of natural numbers, then W is not a finitistic space.Proof Suppose that W is finitistic. Since dim V p > n p , for each positive integer p there exists a finite open cover °U p of V p which has no open refinement of order less than or equal to n p . Then <% = [j °U p is an open cover of W. Since W is finitistic, the open cover °ll of W has an open refinement Y of order not exceeding N for some positive integer N. Since (n p ) pef^ is a strictly increasing sequence, there exists a positive integer q such that n q > N. Since Y q = {V e Y: V <= V q } is an open refinement of ^ of order not exceeding N, we have a contradiction. Hence W is not a finitistic space.The concept of local dimension [4] will be employed in the proof of our theorem. We note that if X is a finitistic normal space then it is weakly paracompact and normal so that [7, Proposition 5.3.4] we have locdimX = dimX. THEOREM. A normal Hausdorff completely finitistic space is finite-dimensional.