I define in my draft article Cauchy filters $\mathcal{X}$ on a uniform space $\nu$ by the formula:

$$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu.$$

$\mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu$ can be defined in elementary terms by the formula $$\mathcal{X}\times^{\mathsf{RLD}}\mathcal{X}\sqsubseteq\nu \Leftrightarrow \forall U \in \operatorname{GR}\nu \exists A \in \mathcal{X} : A \times A \subseteq U$$ (here $\operatorname{GR}\nu$ is the set of entourages of the space $\nu$).

But there can be defined a similar formula for funcoids (a generalization of proximity spaces). A filter $\mathcal{X}$ is Cauchy regarding a funcoid $\nu$ iff

$$\mathcal{X}\ne\bot \wedge \mathcal{X}\times^{\mathsf{FCD}}\mathcal{X}\sqsubseteq\nu.$$

This formula can be rewritten in elementary terms for a proximity space $\nu=(U;\delta)$:

$$\mathcal{X}\times^{\mathsf{FCD}}\mathcal{X}\sqsubseteq\nu \Leftrightarrow \forall P, Q \in U : \left( \forall E \in \mathcal{X} : ( E \cap P \neq \emptyset \wedge E \cap Q \neq \emptyset) \Rightarrow P \mathrel{\delta} Q \right).$$

So, I have defined Cauchy filters for proximity spaces.

My question: Am I the first person who has defined Cauchy filters for proximity spaces? If not, please give me references to definition of Cauchy filters on proximity spaces.

  • $\begingroup$ Hm... maybe all proximity spaces are totally bounded? We need examples or counter-examples $\endgroup$ – porton Feb 28 '14 at 19:55

See Gilman and Jerison's approach to uniform spaces in terms of families of pseudometrics. You can also find the definitions (but not the development) in Willard's book. The upshot is that a proximity space induces a uniformity that is TOTOALLY BOUNDED, in the sense that every pseudometric in the gage uniformity is a totally bounded pseudometric. There is a unique compactification that is obtained as the maximal ideal space of the commutative algebra of all bounded uniformly continuous complex valued functions. The cauchy filters are then defined as in any uniform space: a filter is Cauchy iff it contains an epsilon ball for each positive epsilon. (the point is that the balls come from pseudometrics from the gage uniformity, but the balls need not all come from the same pseudometric.) This was certainly well known by Stone.


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