Localic subspaces and colimits of localic spaces
Tóm tắt
A spectral space is localic if it corresponds to a frame under Stone Duality. This class of spaces was introduced by the author (under the name ’locales’) as the topological version of the classical frame theoretic notion of locales, see Johnstone and also Picado and Pultr). The appropriate class of subspaces of a localic space are the localic subspaces. These are, in particular, spectral subspaces. The following main questions are studied (and answered): Given a spectral subspace of a localic space, how can one recognize whether the subspace is even localic? How can one construct all localic subspaces from particularly simple ones? The set of localic subspaces and the set of spectral subspaces are both inverse frames. The set of localic subspaces is known to be the image of an inverse nucleus on the inverse frame of spectral subspaces. How can the inverse nucleus be described explicitly? Are there any special properties distinguishing this particular inverse nucleus from all others? Colimits of spectral spaces and localic spaces are needed as a tool for the comparison of spectral subspaces and localic subspaces.
Tài liệu tham khảo
Ball R.N., Pultr A., Sichler J.: Tame parts of free summands in coproducts of Priestley spaces. Topology and its Applications 156, 2137–2147 (2009)
Banaschewski B.: Another look at the localic Tychonoff Theorem. Math. Univ. Carolinae 29, 647–656 (1988)
Bourbaki N. (1971) Topologie générale.Chapitres 1 à 4. Hermann, Paris
Cohn, P.M.: Universal Algebra. D. Reidel, Dordrecht (1981)
Dickmann, M., Schwartz, N., Tressl, M.: Spectral Spaces. New Mathematical Monographs, Cambridge University Press (to appear)
Gillman L., Jerison M.: Rings of Continuous Functions. Springer, New York (1976)
Gleason A.M.: Projective topological spaces. Illinois J. Math. 2, 482–489 (1958)
Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)
Grätzer G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)
Herrlich, H., Strecker, G.: Category Theory, 2nd edn. Heldermann, Berlin (1979)
Hochster M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43–60 (1969)
Johnstone P.: Stone Spaces. Cambridge University Press, Cambridge (1986)
Koppelberg S.: Handbook of Boolean Algebras, I. North-Holland, Amsterdam (1989)
Koubek V., Sichler J.: On Priestley duals of products. Cahiers Topologie Géom. Différentielle Catég. 32, 243–256 (1991)
Martinez J., McGovern W.: Free meets and atomic assemblies of frames. Algebra Universalis 62, 153–163 (2009)
Picado J., Pultr A.: Frames and Locales. Topology wthout points. Springer, Basel (2012)
Porter J.R., Woods R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer, New York (1988)
Schwartz N.: Locales as spectral spaces. Algebra Universalis 70, 1–42 (2013)
Schwartz, N.: Spectral reflections of topological spaces. Applied Categorical Structures (to appear)
Schwartz, N.: Limits of locales. In preparation
Simmons, H.: A framework for topology. In: Macintyre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium ‘77, pp. 239–251, North-Holland, Amsterdam (1978)
Stone M.H.: Topological representations of distributive lattices and Brouwerian logics.. Časopis Mat. Fys., Praha 67, 1–25 (1937)
van Mill, J.: An introduction to \(\beta\omega\). In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-theoretic Topology, pp. 503–567. North-Holland, Amsterdam (1984)