On finite-dimensional uniform spaces. II

Abstract

Introduction. The main subject of this paper is the [inductive* dimension δ Ind μX of uniform spaces μX. This is defined similarly to topological dimension Ind, but instead of separation one uses the notion of a set H, arbitrarily small uniform neighborhoods of which uniformly separate given sets A, B. For finite dimensional metric spaces M (i.e. the large dimension Δd M is finite) 8 Ind coincides with the covering dimensions Ad and δd. For general spaces μX we have 8 Ind μX ^ δd μX. For all known examples (including the examples for Δd Φ δd and, in compact spaces, Ind Φ dim) 8 Ind coincides with δd.The last section of the paper concerns the dimension theory of uniformisable spaces; it organizes alternative definitions and formulates problems, giving limited results on some of the problems. Covering dimension dim has been successfully generalized by Smirnov [17]; here we add to Smirnov's theory a generalization of Aleksandrov's theorem characterizing dim by separating ^-tuples of pairs (A i9 Bi) of disjoint closed sets by closed sets C; with empty intersection. The notion of min dim, mentioned in Part I [7], is formally defined: min dim X is the minimum of Id μX over all compatible uniformities μ. Equivalently, it is the minimum of dim Y over spaces Y containing X. The question when min dim X -dim X, i.e. when X cannot be embedded in a space of lower dimension, is stressed. The Lindelof property implies this, but the question is open for metrizable spaces and more generally for spaces admitting a complete uniformity.It is shown that every completely metrizable space can be homeomorphically embedded as a closed set in a countable product of finitedimensional polyhedra. Combined with results of [9] this means that every completely metrizable space is an inverse limit of polyhedra of the same or lower dimension. The question is still open whether a 1-dimensional completely metrizable space can be an inverse limit of discrete spaces.An announcement of the results on 8 Ind appeared in [8],

A simple proof of a theorem of Isbell

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A Simple Proof of a Theorem of Isbell

“…Z is homeomorphic to a closed subset of JXT-oI A(n)|. [2 ] in the same review. The writer regrets these inaccuracies.…”

Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

“…This definition used the notion of a "freeing set." Isbell posed the problem of whether or not δInd and δd coincide in [6] and [7].…”