Constructing the sobrification of an approach space via bicompletion. (English) Zbl 1174.54006
Approach spaces form the missing link in the topology-uniformity-metric triad (R. Lowen). B. Banaschewski, R. Lowen and C. Van Olmen [Topology Appl., 153, No. 16, 3059–3070 (2006; Zbl 1114.54007)] have introduced a notion of sobriety for such approach spaces, modelled after the corresponding concept in the category of topological spaces.
In the note under review based on the classical Császár-Pervin quasi-uniform space, the authors construct a compatible quasi-uniform gauge structure on every approach space \(X\). Via bicompletion of this gauge space of a \(T_0\) approach space \(X\) the sobrification of \(X\) is constructed, retrieving especially the old result of G. C. L. Brümmer and H. P. A. Künzi [Math. Proc. Camb. Philos. Soc. 101, 237–247 (1986; Zbl 0618.54024)] on the sobrification of a \(T_0\) topological space.
In the note under review based on the classical Császár-Pervin quasi-uniform space, the authors construct a compatible quasi-uniform gauge structure on every approach space \(X\). Via bicompletion of this gauge space of a \(T_0\) approach space \(X\) the sobrification of \(X\) is constructed, retrieving especially the old result of G. C. L. Brümmer and H. P. A. Künzi [Math. Proc. Camb. Philos. Soc. 101, 237–247 (1986; Zbl 0618.54024)] on the sobrification of a \(T_0\) topological space.
Reviewer: Heinrich Reitberger (Innsbruck)
MSC:
| 54B30 | Categorical methods in general topology |
| 54E15 | Uniform structures and generalizations |
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
| 54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |
| 18B99 | Special categories |
Keywords:
approach space; quasi-uniform gauge space; totally bounded; sober; bicomplete; Császár-Pervin quasi-uniformityReferences:
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