Semicontinuous Real Functions Research Articles

Selection principle S1 and combinatorics of open covers

O, Ω and Γ denotes the family of all open, open ω- and open γ-covers of a topological space X, respectively. Osh, Ωsh and Γsh denotes the corresponding families of shrinkable covers. Let Ψ be one of the symbols O, Ω or Γ. We introduce a property (Ψ0) of a set of real functions on X. Ψ0(F) is the set of all subsets of a family of real functions F possessing the property (Ψ0). Now, let Φ be one of symbols Ω or Γ. Then for any couple 〈Φ,Ψ〉 different from 〈Ω,O〉, a normal topological space X is an S1(Φsh,Ψ)-space, if and only if the set Cp(X) satisfies S1(Φ0,Ψ0), i.e., if S1(Φ0(Cp(X)),Ψ0(Cp(X))) holds true. Similarly, X is an S1(Φ,Ψ)-space if and only if the set of non-negative upper semicontinuous real functions on X satisfies S1(Φ0,Ψ0).

Normal semicontinuity and the Dedekind completion of pointfree function rings

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.

Baire generalized topological spaces, generalized metric spaces and infinite games

Different generalizations of topological Baire spaces to the case of generalized topological spaces are considered and the properties of such spaces are examined. In particular, these considerations are focused on the relationship between Baire generalized topological spaces and semi-continuous real functions and infinite games. The notion of generalized metric spaces corresponding to generalized topological spaces is introduced as an important tool in this discussion.

Rings of real functions in pointfree topology

This paper deals with the algebra F ( L ) of real functions on a frame L and its subclasses LSC ( L ) and USC ( L ) of, respectively, lower and upper semicontinuous real functions. It is well known that F ( L ) is a lattice-ordered ring; this paper presents explicit formulas for its algebraic operations which allow to conclude about their behaviour in LSC ( L ) and USC ( L ) . As applications, idempotent functions are characterized and previous pointfree results about strict insertion of functions are significantly improved: general pointfree formulations that correspond exactly to the classical strict insertion results of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces are derived. The paper ends with a brief discussion concerning the frames in which every arbitrary real function on the α-dissolution of the frame is continuous.

Insertion of Continuous Real Functions on Spaces, Bispaces, Ordered Spaces and Pointfree Spaces—A Common Root

We characterize normal and extremally disconnected biframes in terms of the insertion of a continuous real function in between given lower and upper semicontinuous real functions and show this to be the common root of several classical and new insertion results concerning topological spaces, bitopological spaces, ordered topological spaces and locales.

Completely normal frames and real-valued functions

Up to now point-free insertion results have been obtained only for semicontinuous real functions. Notably, there is now available a setting for dealing with arbitrary, not necessarily (semi-)continuous, point-free real functions, due to Gutiérrez García, Kubiak and Picado, that gives point-free topology the freedom to deal with general real functions only available before to point-set topology. As a first example of the usefulness of that setting, we apply it to characterize completely normal frames in terms of an insertion result for general real functions. This characterization extends a well-known classical result of T. Kubiak about completely normal spaces. In addition, characterizations of completely normal frames that extend results of H. Simmons for topological spaces are presented. In particular, it follows that complete normality is a lattice-invariant property of spaces, correcting an erroneous conclusion in [Y.-M. Wong, Lattice-invariant properties of topological spaces, Proc. Amer. Math. Soc. 26 (1970) 206–208].

Lower and upper regularizations of frame semicontinuous real functions

As discovered recently, Li and Wang’s 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), 271–281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem.

Localic real functions: A general setting

In pointfree topology the lattice-ordered ring of all continuous real functions on a frame L has not been a part of the lattice of all lower (or upper) semicontinuous real functions on L just because all those continuities involve different domains. This paper demonstrates a framework in which all those continuous and semicontinuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all frame homomorphisms from the frame L ( R ) of reals into S ( L ) , the dual of the co-frame of all sublocales of L . The lattice-ordered ring Frm ( L ( R ) , S ( L ) ) is a pointfree counterpart of the ring R X with X a topological space. We thus have a pointfree analogue of the concept of an arbitrary not necessarily ( semi) continuous real function on L . One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems, proved earlier by these authors, be recast in the new setting.

On the algebraic representation of semicontinuity

The concepts of upper and lower semicontinuity in pointfree topology were introduced and first studied by Li and Wang [Y.-M. Li, G.-J. Wang, Localic Katětov–Tong insertion theorem and localic Tietze extension theorem, Comment. Math. Univ. Carolin. 38 (1997) 801–814]. However Li and Wang’s treatment does not faithfully reflect the original classical notion. In this note, we present algebraic descriptions of upper and lower semicontinuous real functions, in terms of frame homomorphisms, that suggest the right alternative to the definitions of Li and Wang. This fixes the discrepancy between the classical and the pointfree notions and turns out to be the appropriate notion that makes the Katětov–Tong theorem provable in the pointfree context without any restrictions.

Computability on continuous, lower semi-continuous and upper semi-continuous real functions

In this paper we extend computability theory to the spaces of continuous, upper semi-continuous and lower semi-continuous real functions. We apply the framework of TTE, Type-2 Theory of Effectivity, where not only computable elements but also computable functions on the spaces can be considered. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on different intuitive concepts of “effectivity” and prove their equivalence. Computability of several operations on the function spaces is investigated, among others limits, mappings to open sets, images of compact sets and preimages of open sets, maximum and minimum values. The positive results usually show computability in all arguments, negative results usually express discontinuity. Several of the problems have computable but not extensional solutions. Since computable functions map computable elements to computable elements, many previously known results on computability are obtained as simple corollaries.

ON A PROPERTY OF THE SUBDIFFERENTIAL

Semicontinuous real functions are considered. The following property is established for the Dini directional semiderivative and the Dini semidifferential (the subdifferential). If at some point the semiderivative is positive in a convex cone of directions, then arbitrarily close to the point under consideration there exists a point at which the function is subdifferentiable and has a subgradient belonging to the positively dual cone. This result is used in the theory of the Hamilton-Jacobi equations to prove the equivalence of various types of definitions of generalized solutions.

MAPPINGS CONJUGATE TO MULTIVALUED MAPPINGS OF TOPOLOGICAL SPACES, AND THEIR APPLICATION TO DYNAMICAL GAMES

Associated with multivalued mappings of a topological space X into a topological space Y are conjugate mappings of the set of upper (lower) semicontinuous real functions on Y into the set of real functions on X. It. is shown that to compactvalued upper semicontinuous mappings of X into Y there correspond mappings of the set of upper (lower) semicontinuous real functions on Y into the set of upper (lower) semicontinuous real functions on X.The properties of the conjugate mappings are studied, and their applications to dynamical games are considered.