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Zbl 0717.54016
Kopperman, R.D.; Meyer, P.R.
Cardinal invariants of bitopological spaces. (English)
[J] Čas. Pěstování Mat. 114, No.4, 374-380 (1989). ISSN 0528-2195

This paper presents definitions of the analogues of many cardinal functions on bitopological spaces. First are considered the cardinal functions weight, density, cellularity, spread, Lindelöf degree, height, net weight, width, and extent (denoted respectively by w, d, c, s, L, h, n, z, and e). For example, the cellularity of a topological space (X,T), is by definition the smallest cardinality of a family of pairwise disjoint open sets. How should ``bicellularity'' be defined for a bitopological space (X, $T\sb 1$, $T\sb 2)?$ Two obvious cardinal numbers to consider are (1) the product $c((X,T\sb 1))c((X,T\sb 2))$, and the number $c(X,T\sb 1\vee T\sb 2))$, where $T\sb 1\vee T\sb 2$ is the smallest topology on X containing both $T\sb 1$ and $T\sb 2$. To make the definitions, the authors follow the rule that if $\phi$ is one of the cardinal functions from the above list, and if $\phi (T)\le \phi (T')$ whenever T and $T'$ are topologies on a set X with $T\subset T'$, then take for definition $b\phi ((X,T1,T2))=\phi (X,T\sb 1\vee T\sb 2)$; otherwise take for definition $b\phi ((X,T\sb 1,T\sb 2))=\phi (T\sb 1)\phi (T\sb 2)$. Thus, for bicellularity, $bc((X,T\sb 1,T\sb 2))=c(X,T\sb 1\vee T\sb 2)$, and for biweight, $bw((X,T\sb 1,T\sb 2))=w((X,T\sb 1))w((X,T\sb 2))$. Using these definitions, it is shown that the well-known relations among the above cardinal functions extend to their bitopological counterparts. Three further cardinal functions considered are metrization number, uniform weight, and Mrówka number. Definitions for these are given by a different method. The main result states (a) that if $\phi$ is any function from the above list, then $b\phi +bq=bw$ for pairwise $T\sb 0$, pairwise completely regular bitopological spaces, where $bq((X,T\sb 1,T\sb 2))$ is the least cardinality of a base for a quasiuniform space (X,${\cal U})$ such that $T\sb 1$ is the topology generated by ${\cal U}$, and $T\sb 2$ is the topology generated by the dual ${\cal U}\sp*=\{U\sp{-1}:U\in {\cal U}\}$, and (b) $\phi +u=w$ for Tychonoff topological spaces, where u is the smallest cardinality of a uniformity on X which generates the topology on X. Further results are also given.
[J.E.Vaughan]

MSC 2000:: *54E55 Bitopologies
54A25 Cardinality properties of topological spaces
54E15 Uniform structures and generalizations

Keywords: bitopological spaces; weight; density; cellularity; spread; Lindelöf degree; height; net weight; width; extent; metrization number; uniform weight; Mrówka number; quasiuniform space

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