"Approach spaces" and "frames": A further study of the interaction between quantitative and point-free topology

Project Details

Description

The category of approach spaces serves as a quantified counterpart to that of topological spaces: here arbitrary products of metric objects can be formed overlying the topological product but retaining all numerical information present in the factors. The recent monograph published with Springer [23] illustrates many applications in analysis, theoretical computer science and probability. A first monograph on the topic published with Oxford University Press [22] received a featured review with Math. Rev. of the AMS. The category of frames or its dual, the category of locales, are the appropriate setting to treat topological concepts in a universally-algebraic way, without reference to points. Frames naturally arise as spectra of commutative rings, but "having to give up points" also happens in other contexts e.g. in non-commutative geometry. Apart from the broader scope, studying topology in this pointfree setting gives more insight, often avoiding choice principles or allowing a constructive approach. Strong ties between pointfree topology and approach theory were discovered in [6,7,8], a very fruitful link bringing more insight (see [23]). More recently, in [23], it has been shown that also a good counterpart to upper semicontinuity exists in approach theory, hinting at an even more intimate link between quantified and pointfree topology. The aim of this project is to further investigate this interesting interaction and exploit it to obtain new results in both settings.
AcronymFWOKN288
StatusFinished
Effective start/end date1/01/1731/12/17

Flemish discipline codes

  • General mathematics
  • General mathematics

Keywords

  • approach spaces