# zbMATH — the first resource for mathematics

Categories of diametric frames. (English) Zbl 0683.54008
This paper is concerned with extending metric structure to pointless spaces (frames or locales). The distance function is replaced by a “diameter” function from the frame to the non-negative reals satisfying certain natural properties (monotonicity, subadditivity). Two additional properties are postulated to obtain a well-behaved theory; the resulting structures are called diametric frames. Various categories of diametric frames are considered, and the adjoint relationship between the open-set functor and the spectrum functor established. The paper closes with a consideration of limits and colimits in this setting.
Reviewer: J.D.Lawson

##### MSC:
 54A05 Topological spaces and generalizations (closure spaces, etc.) 18B30 Categories of topological spaces and continuous mappings (MSC2010) 54H12 Topological lattices, etc. (topological aspects) 06B30 Topological lattices
##### Keywords:
pointless spaces; frames; locales; diametric frames; limits; colimits
Full Text:
##### References:
 [1] Pultr, Proc. 12th Winter School 6 pp 247– (1984) [2] Pultr, Comment. Math. Univ. Carolin 25 pp 105– (1984) [3] Pultr, Comment. Math. Univ. Carolin 25 pp 91– (1984) [4] Isbell, Math. Scand 31 pp 5– (1972) · Zbl 0246.54028 [5] Johnstone, Fund. Math 113 pp 21– (1981) [6] Johnstone, Stone Spaces (1982) [7] K???, Cahiers Topologie G?om. Differentielle Cat?goriques XXVII 1 pp 19– (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.