Localic maps constructed from open and closed parts

Document Type: Research Paper

Authors

1 Department of Applied Mathematics and ITI, MFF, Charles University, Malostransk'e n'am. 24, 11800 Praha 1, Czech Republic.

2 CMUC, Department of Mathematics, University of Coimbra, Apar-ta-do 3008, 3001-501 Coimbra, Portugal.

Abstract

Assembling a localic map $f\colon L\to M$ from localic maps $f_i\colon S_i\to M$, $i\in J$, defined on closed resp. open sublocales $(J$ finite in the closed case$)$ follows the same rules as in the classical case. The corresponding classical facts immediately follow from the behavior of  preimages but for obvious reasons such a proof cannot be imitated in the point-free context. Instead,  we present  simple proofs based on categorical reasoning. There are some related aspects of localic preimages that are of interest, though. They are investigated in the second half of the paper.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his 90th birthday

Keywords


[1] J. R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972), 5-32.
[2] P. T. Johnstone,  "Stone Spaces", Cambridge Univ. Press, Cambridge, 1982.
[3] J. L. Kelley, "General Topology", Van Nostrand, 1955.
[4] S. Mac Lane, "Categories for the Working Mathematician", Springer-Verlag, New York, 1971.
[5] J. Picado and A. Pultr, "Locales treated mostly in a covariant way", Textos de Matem'{a}tica, Vol. 41, University of Coimbra, 2008.
[6] J. Picado and A. Pultr,  "Frames and Locales: topology without points", Frontiers in Mathematics, Vol. 28, Springer, Basel, 2012.
[7] T. Plewe, Quotient maps of locales, Appl. Categ. Structures 8 (2000), 17-44.