Abstract
The concept of sobriety of ordinary topological spaces has been around since at least the 1970’s. Sober topological spaces were introduced by Grothendieck et al ([1] IV 4.2.1) and independently by T. Blanksma [3]. Since then various authors studied this axiom, e.g. R-E. Hoffmann ([5] and [6]) and P. T. Johnstone [10]. It was pointed out by [21] and [25] that this axiom is also of importance to computer science, domain theory in particular. Smyth argues that “computationally reasonable spaces are sober”. There are several reasons for this, one being the possibility of, in the case of sobriety, moving from frame maps between frames to continuous maps between topological spaces; in other words, continuous maps being categorically (dually) equivalent to frame maps requires sobriety (see e.g. [11], Theorems 3.3 and 3.4). Another is that the Scott topology on continuous posets are highly non-Hausdorff but sober. See e.g. [10]. (The Scott topology on a two point set is the Sierpinski space).
This work was supported by a grant from the NRF (South Africa). I wish to thank A. K. Srivastava for his most valuable input and stimulation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Artin, A. Grothendieck, J. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics 269, Springer-Verlag (Berlin/Heidelberg/New York), 1972.
S. F. Barger, Heredity sobriety and the cardinality of T 1 topologies on countable sets,Quaestiones Mathematicae 20(3)(1997) 117–126.
T. Blanksma, Lattice characterizations and compactifications, Doctoral Dissertation, Rijksuniversiteit te Utrecht 1968, MR 37, p. 5851.
G. Gierz et al, A Compendium Of Continuous Lattices, Springer-Verlag (Berlin/Heidelberg/New York), 1980.
R.-E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977), 365–380.
R.-E. Hoffmann, Sobrification of partially ordered sets, Semigroup Forum 17 (1979), 123–138.
U. Höhle, Uppersemicontinuous fuzzy sets and applications, J. Math. Anal. Appl. 78 (1978), 659–673.
U. Höhle, Locales and L-topologies,Mathematik-Arbeitspapiere 48(1997), 223–250,Universität Bremen.
U. Höhle, S. E. Rodabaugh, eds, Mathematics Of Fuzzy Sets: Logic, Topology, And Measure Theory, The Handbooks of Fuzzy Sets Series, Volume 3, Kluwer Academic Publishers (Boston/Dordrecht/London), 1999.
P. T. Johnstone, Stone Spaces, Cambridge University Press (Cambridge), 1982.
W. Kotzé, Lattice morphisms, sobriety, and Urysohn lemmas,Chapter 10 in [20], 257–274.
W. Kotzé, Fuzzy sobriety and fuzzy Hausdorff, Quaestiones Mathematicae 20 (1997), 415–422.
W. Kotzé, Sobriety and semi-sobriety of L-topological spaces, Quaestiones Mathematicae 24 (2001), 549–554.
T. Kubiak, On L-Tychonoff spaces, Fuzzy Sets and Systems 73 (1995), 25–53.
T. Kubiak, A. P. Sostak, Lower set-valued fuzzy topologies Quaestiones Mathematicae 20(3)(1997), 423–429.
S. E. Rodabaugh, A point-set lattice theoretic framework T for topology which contains LOC as a subcategory of singleton spaces and in which there are general classes of Stone Representations and compactification theorems, Youngstown State University Central Printing Office (first edition February 1986, second edition April 1987 ), Youngstown, Ohio.
S. E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets and Systems 40 (1991), 297–345.
S. E. Rodabaugh, Applications of localic separation axioms, compactness axioms, representations, and compactifications to poslat topological spaces, Fuzzy Sets and Systems 73 (1995), 55–87.
S. E. Rodabaugh, Separation axioms: representation theorems, compactness, and compactifications,Chapter 7 in [9], 481–552.
S. E. Rodabaugh, E. P. Klement, U. Höhle, eds, Application Of Category To Fuzzy Sets, Theory and Decision Library—Series B: Mathematical and Statistical Methods, Volume 14, Kluwer Academic Publishers (Boston/Dordrecht/London), 1992.
M. B. Smyth, Topology, Handbook of Logic in Computer Science, Vol. I, Clarendon Press (Oxford), 1992, pp. 641–761.
A. P. ostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, sr. II, 11 (1985), 89–103.
A. P. Sostak, On compactness and connectedness degrees of fuzzy sets in fuzzy topological spaces. General Topology and related Modern Analysis and Algebra. Proceedings of the V-th Prague Topological Symposium, Prague 1986, 519–523; Heldermann Verlag (Berlin), 1988.
A. K. Srivastava, A. S. Khastgir, On fuzzy sobriety, Information Sciences 110 (1998), 195–205.
S. Vickers, Topology Via Logic, Cambridge Tracts in Theoretical Computer Science 5, Cambridge University Press, 1989.
Mingsheng Ying, A new approach for fuzzy topology (I),Fuzzy Sets and Systems 39(1991), 303–321.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Kotzé, W. (2003). Lifting Of Sobriety Concepts With Particular Reference To (L, M)-Topological Spaces. In: Rodabaugh, S.E., Klement, E.P. (eds) Topological and Algebraic Structures in Fuzzy Sets. Trends in Logic, vol 20. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0231-7_18
Download citation
DOI: https://doi.org/10.1007/978-94-017-0231-7_18
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6378-6
Online ISBN: 978-94-017-0231-7
eBook Packages: Springer Book Archive