Realcompactifications of frames

Bibliography: leaves 53-57.The first notion of realcompactness in frames was introduced by Reynolds [1979], and it was shown by Madden and Vermeer [1986] that this coincides with the Lindelof property. My thesis advisor suggested that more general realcompactifications of a frame L could be constructed by considering regular sub σ-frames which join generate L. This was motivated by the fact that the Alexandroff bases, which are used to construct the Wallman realcompactifications of a space X, are, as shown by Gilmour, simply the regular sub σ-frames of the frame of open sets of X. The key definition of realcompactness needed here is due to Schlitt [1990] and it is his construction of the universal realcompactification that we modify in order to obtain the Wallman realcompactifications

E-compactness in pointfree topology

Bibliography: leaves 100-107.The main purpose of this thesis is to develop a point-free notion of E-compactness. Our approach follows that of Banascheski and Gilmour in [17]. Any regular frame E has a fine nearness and hence induces a nearness on an E-regular frame L. We show that the frame L is complete with respect this nearness iff L is a closed quotient of a copower of E. This resembles the classical definition, but it is not a conservative definition: There are spaces that may be embedded as closed subspaces of powers of a space E, but their frame of opens are not closed quotients of copowers of the frame of opens of E. A conservative definition of E-compactness is obtained by considering Cauchy completeness with respect to this nearness. Another central notion in the thesis is that of K-Lindelöf frames, a generalisation of Lindelöf frames introduced by J.J. Madden [59]. In the last chapter we investigate the interesting relationship between the completely regular K-Lindelöf frames and the K-compact frames

A few points on Pointfree pseudocompactness

We present several characterizations of completely regular pseudocompact frames. The first is an extension to frames of characterizations of completely regular pseudocompact spaces given by Väänänen. We follow with an embedding-type characterization stating that a completely regular frame is pseudocompact if and only if it is a P-quotient of its Stone-Čech compactification. We then give a characterization in terms of ideals in the cozero parts of the frames concerned. This characterization seems to be new and its spatial counterpart does not seem to have been observed before. We also define relatively pseudocompact quotients, and show that a necessary and sufficient condition for a completely regular frame to be pseudocompact is that it be relatively pseudocompact in its Hewitt realcompactification. Consequently a proof of a result of Banaschewski and Gilmour that a completely regular frame is pseudocompact if and only if its Hewitt realcompactification is compact, is presented without the invocation of the Boolean Ultrafilter Theorem

Realcompact Alexandroff spaces and regular σ-frames

Bibliography: pages 96-103.In the early 1940's, A.D. Alexandroff [1940), [1941) and [1943] introduced a concept of space, more general than topological space, in order to obtain a simple connection between a space and the system of real-valued functions defined on it. Such a connection aided the investigation of the relationships between the linear functionals on these systems of functions and the additive set functions defined on the space. The Alexandroff spaces of this thesis are what Alexandroff himself called the completely normal spaces and what H. Gordon [1971) called the zero-set spaces

Pointfree pseudocompactness revisited

We give several internal and external characterizations of pseudocompactness in frames which extend (and transcend) analogous characterizations in topological spaces. In the case of internal characterizations we do not make reference (explicitly or implicitly) to the reals

On the Menger and almost Menger properties in locales

[EN] 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.The second-named author acknowledges funding from the National Research Foundation of South Africa under Grant 113829.Bayih, T.; Dube, T.; Ighedo, O. (2021). On the Menger and almost Menger properties in locales. Applied General Topology. 22(1):199-221. https://doi.org/10.4995/agt.2021.14915OJS199221221R. 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/dm412-0-1B. Banaschewski and C. Gilmour, Pseudocompactness and the cozero part of a frame, Comment. Math. Univ. Carolin. 37 (1996), 579-589B. Banaschewski and A. Pultr, Variants of openness, Appl. Categ. Structures 2 (1994), 331-350. https://doi.org/10.1007/BF00873038M. Bonanzinga, F. Cammaroto and M. Matveev, Projective versions of selection principles, Topology Appl. 157 (2010), 874-893. https://doi.org/10.1016/j.topol.2009.12.004C. 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/10.2989/16073606.2013.779986T. 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/s10485-013-9341-8M. 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.2018.06.007J. 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-zW. 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/S0002-9939-2012-11329-2P. 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/FIL1501099KJ. Madden and J. Vermeer, Lindelöf locales and realcompactness, Math. Proc. Camb. Phil. Soc. 99 (1986), 473-480. https://doi.org/10.1017/S0305004100064410J. Picado and A. Pultr, Frames and Locales: topology without points, Frontiers in Mathematics, Springer, Basel, 2012. https://doi.org/10.1007/978-3-0348-0154-6J. 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)00100-1V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41-74. https://doi.org/10.24033/bsmf.1498Y.-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/PIM150513031SJ. J. C. Vermeulen, Proper maps of locales, J. Pure Appl. Algebra 92 (1994), 79-107. https://doi.org/10.1016/0022-4049(94)90047-

Variants of P-frames and associated rings

We study variants of P-frames and associated rings, which can be viewed as natural generalizations of the classical variants of P-spaces and associated rings. To be more precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either maximal or minimal. For a completely regular frame L, if the ring RL of real-valued continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented frame. These frames are less restricted than the cozero complemented frames. Using these frames we study some properties of what are called quasi m-spaces, and observe that the property of being a quasi m-space is inherited by cozero subspaces, dense z- embedded subspaces, and regular-closed subspaces among normal quasi m-space. M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal associated with I is either maximal or minimal. If all points of L are quasi P-points, we say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi P-space if and only if the frame OX is a quasi P-frame. We characterize these frames in terms of cozero elements, and, among cozero complemented frames, give a su cient condition for a frame to be a quasi P-frame. A Tychono space X is called a weak almost P-space if for every two zero-sets E and F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H. We present the pointfree version of weakly almost P-spaces. We de ne weakly regular rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring RL of real-valued continuous functions on L is weakly regular. We introduce the notions of boundary frames and boundary rings, and use them to give another ring-theoretic characterization of boundary spaces. We show that X is a boundary space if and only if C(X) is a boundary ring. A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces each of which is an F-space is said to be nitely an F-space. Among normal spaces, S. Larson gave a characterization of these spaces in terms of properties of function rings C(X). By extending this notion to frames, we show that the normality restriction can actually be dropped, even in spaces, and thus we sharpen Larson's result.MathematicsD. Phil. (Mathematics