Rocky Mountain Journal of Mathematics

The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications

Carmelo A. Finocchiaro, Marco Fontana, and Dario Spirito

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Given an arbitrary spectral space $X$, we consider the set $\mathcal{X} (X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on $\mathcal{X} (X)$ and, after observing that it coincides the upper Vietoris topology, we prove that $\mathcal{X} (X)$ is itself a spectral space, that this construction is functorial, and that $\mathcal{X} (X)$ provides an extension of $X$ in a more ``complete'' spectral space. Among the applications, we show that, starting from an integral domain $D$, $\mathcal{X} (Spec (D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$.

Article information

Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1551-1583.

First available in Project Euclid: 19 October 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A10 13A15: Ideals; multiplicative ideal theory 13B10: Morphisms 13G05: Integral domains 14A05: Relevant commutative algebra [See also 13-XX] 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54F65: Topological characterizations of particular spaces

Spectral space spectral map Zariski topology constructible topology inverse topology hull-kernel topology stably compact space Smyth powerdomain co-compact topology de Groot duality upper Vietoris topology Scott topology closure operation semistar operation radical ideal ultrafilter topology


Finocchiaro, Carmelo A.; Fontana, Marco; Spirito, Dario. The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications. Rocky Mountain J. Math. 48 (2018), no. 5, 1551--1583. doi:10.1216/RMJ-2018-48-5-1551.

Export citation


  • D.D. Anderson, Star-operations induced by overrings, Comm. Algebra 16 (1988), 2535–2553.
  • D.F. Anderson and D.D. Anderson, Examples of star operations on integral domains, Comm. Algebra 18 (1990), 1621–1643.
  • M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, 1969.
  • G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia and A. Kurz, Bitopological duality for distributive lattices and Heyting algebras, Math. Struct. Comp. Sci. 20 (2010), 359–393.
  • C. Chevalley and H. Cartan, Schémas normaux; morphismes; ensembles constructibles, Sem. Cartan 8 (1955)–(1956), 1–10.
  • W.H. Cornish, On H. Priestley's dual of the category of bounded distributive lattices, Mat. Vesnik 12 (1975), 329–332.
  • D.E. Dobbs and M. Fontana, Kronecker function rings and abstract Riemann surfaces, J. Algebra 99 (1986), 263–274.
  • N. Epstein, A guide to closure operations in commutative algebra, Progr. Commut. Alg. 2 (2012), 1–37.
  • ––––, Semistar operations and standard closure operations, Comm. Algebra 43 (2015), 325–336.
  • C.A. Finocchiaro, Spectral spaces and ultrafilters, Comm. Algebra 42 (2014), 1496–1508.
  • C.A. Finocchiaro, M. Fontana, and K.A. Loper, The constructible topology on spaces of valuation domains, Trans. Amer. Math. Soc. 365 (2013), 6199–6216.
  • C.A. Finocchiaro, M. Fontana and D. Spirito, A topological version of Hilbert's Nullstellensatz, J. Algebra 461 (2016), 25–41.
  • ––––, Spectral spaces of semistar operations, J. Pure Appl. Alg. 220 (2016), 2897–2913.
  • ––––, New distinguished classes of spectral spaces: A survey, in Multiplicative ideal theory and factorization theory–Commutative and non-commutative perspectives, S. Chapman, M. Fontana, A. Geroldinger, et al., eds., Springer Verlag, New York, 2016.
  • C.A. Finocchiaro and D. Spirito, Some topological considerations on semistar operations, J. Algebra 409 (2014), 199–218.
  • M. Fontana, Topologically defined classes of commutative rings, Ann. Mat. Pura Appl. 123 (1980), 331–355.
  • M. Fontana and K.A. Loper, Kronecker function rings: A general approach, in Ideal theoretic methods in commutative algebra, Lect. Notes Pure Appl. Math. 220 (2001), 189–205.
  • ––––, Cancellation properties in ideal systems: A classification of e.a.b. semistar operations, J. Pure Appl. Alg. 213 (2009), 2095–2103.
  • G. Gierz, K.H. Hofmann, K. Keimel, et al., Continuous lattices and domains, in Encycl. Math. Appl. 93, Cambridge University Press, Cambridge, 2003.
  • R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
  • A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique I, Springer, Berlin, 1970.
  • F. Halter-Koch, Ideal systems, An introduction to multiplicative ideal theory, Mono. Text. Pure Appl. Math. 211 (1998).
  • ––––, Localizing systems, module systems, and semistar operations, J. Algebra 238 (2001), 723–761.
  • ––––, Kronecker function rings and generalized integral closures, Comm. Algebra 31 (2003), 45–59.
  • ––––, Lorenzen monoids: A multiplicative approach to Kronecker function rings, Comm. Algebra 43 (2015), 3–22.
  • M. Henriksen and R. Kopperman, A general theory of structure spaces with applications to spaces of prime ideals, Algebra Univ. 28 (1991), 349–376.
  • O. Heubo-Kwegna, Kronecker function rings of transcendental field extensions, Comm. Algebra 38 (2010), 2701–2719.
  • M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60.
  • W. Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche I–II, Math. Z. 41 (1936), 665–679.
  • J. Lawson, Stably compact spaces, Math. Struct. Comp. Sci. 21 (2011), 125–169.
  • E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 15–18.
  • A. Okabe and R. Matsuda, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994), 1–21.
  • B. Olberding, Noetherian spaces of integrally closed rings with an application to intersections of valuation rings, Comm. Algebra 38 (2010), 3318–3332.
  • ––––, Intersections of valuation overrings of two-dimensional Noetherian domains, in Commutative algebra, Noetherian and non-Noetherian perspectives, Springer, New York, 2011.
  • ––––, Affine schemes and topological closures in the Zariski-Riemann space of valuation rings, J. Pure Appl. Algebra 219 (2015), 1720–1741.
  • ––––, Topological aspects of irredundant intersections of ideals and valuation rings, in Multiplicative ideal theory and factorization theory, Springer, New York, 2016.
  • H.A. Priestley, Representation of distributive lattices by means of ordered Stone spaces, Bull. Lond. Math. Soc. 2 (1970), 186–190.
  • ––––, Ordered topological spaces and the representation of distributive lattices, Proc. Lond. Math. Soc. 24 (1972), 507–530.
  • N. Schwartz and M. Tressl, Elementary properties of minimal and maximal points in Zariski spectra, J. Algebra 323 (2010), 698–728.
  • H. Simmons, A couple of triples, Topol. Appl. 13 (1982), 201–223.
  • M.B. Smyth, Power domains and predicate transformers: A topological view, in Automata, languages and programming, Lect. Notes Comp. Sci. 154 (1983), 662–675.
  • M.H. Stone, The theory of representation for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111.
  • ––––, Topological representations of distributive lattices and Brouwerian logics, Casopis Pest. Mat. Fys. 67 (1937), 1–25.
  • L. Vietoris, Bereiche zweiter Ordnung, Monatsh. Math. Phys. 32 (1922), 258–280.