© 1996, Topology Atlas

A uniform space is said to be fine provided that its uniformity is the finest one compatible with the topology (the existence of such a uniformity and its fundamental properties may be found in Isbell's book [2]). In paracompact spaces, every open covering is uniform in the fine uniformity. A recurrent problem about uniform spaces is to see whether a given uniformity is fine. Isiwata [3] and Atsuji [1] solved this problem in metric spaces. Their characterizations can be sumarized as follows:

The following conditions on a metric space X are equivalent:
1) every open covering is uniform;
2) every open covering consisting of two elements is uniform;
3) every continuous real-valued function is uniformly continuous;
4) the subset K of limit points is compact and, for every uniform covering U, the subspace X \ St(K,U) is uniformly discrete.

"1) iff 2)" holds also in omega_mu-metric spaces [4]. Condition 2) can be partially generalized to arbitrary uniform spaces: every open covering is uniform provided that every open covering of power not exceeding the uniform weight is uniform [4]. A uniform subspace of a fine space is said to be subfine. The main result about subfine spaces is their coincidence with locally fine spaces [5].

[1] M. Atsuji, "Uniform Continuity of Continuous Functions of Metric Spaces", Pacific J. Math., 8 (1958), 11-16

[2] J.R.Isbell, "Uniform Spaces", Mathematical Surveys nr 12 AMS, Providence, Rhod Island, (1964)

[3] T. Isiwata, "On Uniform Continuity of C(X)" (Japanese), Sugaku Kenkiu Roku of Tokyo Kyoiku Daigaku, 2 (1955), 36-45

[4] U. Marconi, "Some Conditions under which a Uniform Space is Fine", CMUC, 34,3 (1993), 543-547

[5] J. Pelant, "Locally Fine Uniformities and Normal Covers", Czech. Math. Jour., 37/112 (1987), 181-187.