On the Menger and almost Menger properties in locales

Tilahun Bayih, Themba Dube, Oghenetega Ighedo


The Menger and the almost Menger properties are extended to locales. Regarding the former, the extension is conservative (meaning that a space is Menger if and only if it is Menger as a locale), and the latter is conservative for sober TD-spaces. Non-spatial Menger (and hence almost Menger) locales do exist, so that the extensions genuinely transcend the topological notions. We also consider projectively Menger locales, and show that, as in spaces, a locale is Menger precisely when it is Lindelöf and projectively Menger. Transference of these properties along localic maps (via direct image or pullback) is considered.


Menger; almost Menger; frame; locale; sublocale; spectrum of a frame

Subject classification

06D22; 54C05; 54D20.

Full Text:



R. N. Ball and J. Walters-Wayland, C- and C*-quotients in pointfree topology, Dissert. Math. (Rozprawy Mat.) 412 (2002), 1-62. https://doi.org/10.4064/dm412-0-1

B. Banaschewski and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 579-589

B. Banaschewski and A. Pultr, Variants of openness, Appl. Categ. Structures 2 (1994), 331-350. https://doi.org/10.1007/BF00873038

M. Bonanzinga, F. Cammaroto and M. Matveev, Projective versions of selection principles, Topology Appl. 157 (2010), 874-893. https://doi.org/10.1016/j.topol.2009.12.004

C. H. Dowker and D. Strauss, Paracompact frames and closed maps, in: Symposia Mathematica, Vol. XVI, pp. 93-116 (Convegno sulla Topologia Insiemistica e Generale, INDAM, Rome, 1973) Academic Press, London, 1975.

C. H. Dowker and D. Strauss, Sums in the category of frames, Houston J. Math. 3 (1976), 17-32.

T. Dube, M. M. Mugochi and I. Naidoo, Cech completeness in pointfree topology, Quaest. Math. 37 (2014), 49-65. https://doi.org/10.2989/16073606.2013.779986

T. Dube, I. Naidoo and C. N. Ncube, Isocompactness in the category of locales, Appl. Categ. Structures 22 (2014), 727-739. https://doi.org/10.1007/s10485-013-9341-8

M. J. Ferreira, J. Picado and S. M. Pinto, Remainders in pointfree topology, Topology Appl. 245 (2018), 21-45. https://doi.org/10.1016/j.topol.2018.06.007

J. Gutiérrez García, I. Mozo Carollo and J. Picado, Normal semicontinuity and the Dedekind completion of pointfree function rings, Algebra Universalis 75 (2016), 301-330. https://doi.org/10.1007/s00012-016-0378-z

W. He and M. Luo, Completely regular proper reflection of locales over a given locale, Proc. Amer. Math. Soc. 141 (2013), 403-408. https://doi.org/10.1090/S0002-9939-2012-11329-2

P. T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982.

D. Kocev, Menger-type covering properties of topological spaces, Filomat 29 (2015), 99-106. https://doi.org/10.2298/FIL1501099K

J. Madden and J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480. https://doi.org/10.1017/S0305004100064410

J. Picado and A. Pultr, Frames and Locales: topology without points, Frontiers in Mathematics, Springer, Basel, 2012. https://doi.org/10.1007/978-3-0348-0154-6

J. Picado and A. Pultr, Axiom $T_D$ and the Simmons sublocale theorem, Comment. Math. Univ. Carolin. 60 (2019), 701-715.

J. Picado and A. Pultr, Notes on point-free topology, manuscript.

T. Plewe, Sublocale lattices, J. Pure Appl. Algebra 168 (2002), 309-326. https://doi.org/10.1016/S0022-4049(01)00100-1

V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41-74. https://doi.org/10.24033/bsmf.1498

Y.-K. Song, Some remarks on almost Menger spaces and weakly Menger spaces, Publ. Inst. Math. (Beograd) (N.S.) 112 (2015), 193-198. https://doi.org/10.2298/PIM150513031S

J. J. C. Vermeulen, Proper maps of locales, J. Pure Appl. Algebra 92 (1994), 79-107. https://doi.org/10.1016/0022-4049(94)90047-7

Abstract Views

Metrics Loading ...

Metrics powered by PLOS ALM

Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt