# On the Pettis-Johnstone theorem for localic groups

@inproceedings{Chen2021OnTP, title={On the Pettis-Johnstone theorem for localic groups}, author={Ruiyuan Chen}, year={2021} }

We explain how Johnstone’s 1989 proof of the closed subgroup theorem for localic groups can be viewed as a point-free version of Pettis’s theorem for Baire topological groups. We then use it to derive localic versions of the open mapping theorem and automatic continuity of Borel homomorphisms, as well as the non-existence of binary coproducts of complete Boolean algebras.

#### References

SHOWING 1-10 OF 19 REFERENCES

On the completeness of localic groups

- Mathematics
- 1999

The main purpose of this paper is to show that any localic group is complete in its two-sided uniformity, settling a problem open since work began in this area a decade ago. In addition, a number of… Expand

Unsurprising results on localic groups

- Mathematics
- 1990

Abstract If A and B are subsets of a topological group, one of which is open, then AB is open. A ‘pointless’ proof of the corresponding result for localic groups is given. A fundamental tool is Lemma… Expand

On the non-existence of free complete Boolean algebras

- Mathematics
- 1964

Rieger asked in 1951 if there exists a free complete Boolean
algebra on ω complete generators. Crawley and Dean proved in
1955 that there does not exist a free complete lattice on three complete… Expand

A constructive “Closed subgroup theorem” for localic groups and groupoids

- Mathematics
- 1989

© Andrée C. Ehresmann et les auteurs, 1989, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les… Expand

Borel and analytic sets in locales.

- Mathematics
- 2020

We systematically develop analogs of basic concepts from classical descriptive set theory in the context of pointless topology. Our starting point is to take the elements of the free complete Boolean… Expand

First steps in descriptive theory of locales

- Mathematics
- 1991

F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is F, and G. . This extends to arbitrary metrizable locales when "F," and "G."… Expand

New proof of a theorem of Gaifman and Hales

- Mathematics
- 1966

The sets U{n, ƒ) for n <co form a neighborhood basis of ƒ. Let (B be the algebra of regular open sets of X. (An open set U is regular if £/=int(cl U): Here int E is the interior of E and cl E is the… Expand

Frames and Locales - Topology without points

- Mathematics, Computer Science
- Frontiers in mathematics
- 2011

This chapter discusses the structure of localic morphisms in Spaces and lattices of open sets, which describes the construction of frames and locales in the real world. Expand

Invariant Descriptive Set Theory

- Mathematics
- 2008

Preface Polish Group Actions Preliminaries Polish spaces The universal Urysohn space Borel sets and Borel functions Standard Borel spaces The effective hierarchy Analytic sets and SIGMA 1/1 sets… Expand