Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ?-ideals contained in the maximal ?-ideal M _x  = {f ? C(X):f(x) = 0}. The space X has finite rank if there is an n ? N such that every point x ? X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ? X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.     

Images and Open Subspaces of SV Spaces

A topological space is finitely an F-space if its Stone–?ech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). This article studies the images under open continuous functions and the open subspaces of spaces that are finitely an F-space or are SV. It is shown that an open continuous image of a compact space that is finitely an F-space is finitely an F-space and an open continuous image of certain SV spaces is SV. Also, it is shown cozerosets, but not necessarily open sets, of SV spaces are SV spaces and a similar situation holds for spaces that are finitely an F-space.     

On resolvable primal spaces

A topological space is called resolvable if it is a union of two disjoint dense subsets, and is n-resolvable if it is a union of n mutually disjoint dense subsets. Clearly a resolvable space has no isolated points. If f is a selfmap on X, the sets A?X with f (A)?A are the closed sets of an Alexandroff topology called the primal topology 𝒫(f ) associated with f. We investigate resolvability for primal spaces (X, 𝒫(f)). Our main result is that an Alexandroff space is resolvable if and only if it has no isolated points. Moreover, n-resolvability and other related concepts are investigated for primal spaces.     

Splittability for Partially Ordered Sets

A topological space X is said to be splittable over a class P of spaces if for every AX there exists continuous f:XYP such that f(A)f(XA) is empty. A class P of topological spaces is said to be a splittability class if the spaces splittable over P are precisely the members of P. We extend the notion of splittability to partially ordered sets and consider splittability over some elementary posets. We identify precisely which subsets of a poset can be split along over an n-point chain. Using these results it is shown that the union of two splittability classes need not be a splittability class and a necessary condition for P to be a splittability class is given.     

Rings of quotients of f-rings by gabriel filters of ideals

The article examines the role of Gabriel filters of ideals in the ontext of semiprime f-rings. It is shown that for every 2-convex semiprime f-ring Aand every multiplicative filter B of dense ideals the ring of quotients of A by B, namely the direct limit of the Hom _A (I, A) over all I∈ B, is an l-subring of QA, the maximum ring of quotients. Relative to the category of all commutative rings with identity, it is shown that for every 2-convex semiprime f-ring A qA, the classical ring of quotients, is the largest flat epimorphic extension of A. If Ais also a Prüfer ring then it follows that every extension of Ain qA is of the form S ^-1A for a suitable multiplicative subset S. The paper also examines when a Utumi ring of quotients of a semiprime f-ring is obtained from a Gabriel filter. For a ring of continuous functions C(X), with Xcompact, this is so for each C(U) and C ^*(U), when Uis dense open, but not for an arbitrary direct limit of C(U),taken over a filter base of dense open sets. In conclusion, it is shown that, for a complemented semiprime f-ring A, the ideals of Awhich are torsion radicals with respect to some hereditary torsion theory are precisely the intersections of minimal prime ideals of A.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

A note on the rank of semigroups

The rank of a semigroup $\mathcal{A}The rank of a semigroup A\mathcal{A} of functions from a finite set X to X is the minimum of |f(X)| over f ? Af\in \mathcal{A}. Given a finite set X and a subset Y of X, we show that if A\mathcal{A} is a semigroup of functions from X to X and ℬ a transitive semigroup of functions from Y to Y, then the rank of A\mathcal{A} divides that of ℬ provided that f(X)⊆Y for some f ? Af\in \mathcal{A} and that each function in ℬ is the restriction of a function in A\mathcal{A} to Y. To prove this, we generalize a result of Friedman which says that one can partition Y into q subsets of equal weight where q is the rank of ℬ. When one extends a transitive automaton by adding new states and letters, a similar condition guarantees that the rank of the extension divides the original rank.     

On scatteredly continuous maps between topological spaces

A map f:X→Y between topological spaces is defined to be scatteredly continuous if for each subspace AX the restriction f|A has a point of continuity. We show that for a function f:X→Y from a perfectly paracompact hereditarily Baire Preiss–Simon space X into a regular space Y the scattered continuity of f is equivalent to (i) the weak discontinuity (for each subset AX the set D(f|A) of discontinuity points of f|A is nowhere dense in A), (ii) the piecewise continuity (X can be written as a countable union of closed subsets on which f is continuous), (iii) the G_δ-measurability (the preimage of each open set is of type G_δ). Also under Martin Axiom, we construct a G_δ-measurable map f:X→Y between metrizable separable spaces, which is not piecewise continuous. This answers an old question of V. Vinokurov.     

Homotopy Properties of the Space If(X) of Idempotent Probability Measures

A subspace I_f(X) of the space of idempotent probability measures on a given compact space X is constructed. It is proved that if the initial compact space X is contractible, then I_f(X) is a contractible compact space as well. It is shown that the shapes of the compact spaces X and I_f(X) are equal. It is also proved that, given a compact space X, the compact space I_f(X) is an absolute neighborhood retract if and only if so is X.

     

α-Baer rings and some related concepts via C(X)

We call a commutative ring R an F IN -ring (resp., F SA-ring) if for any two finitely generated I, J ?R we have Ann(I)+Ann(J )=Ann(I∩J ) (resp., there is K ? R such that Ann(I)+Ann(J )=Ann(K)). Moreover, we extend this concepts to αIN -rings and αSA-rings where α is a cardinal number. The class of F SA-rings includes the class of all SA-rings (hence all IN -rings) and all P P -rings (hence all Baer-rings). In this paper, after giving some properties of αSA-rings, we prove that a reduced ring R is αSA if and only if it is an αIN -ring. Consequently, C(X) is an F SA-ring if and only if C(X) is an F IN -ring and equivalently X is an F -space. Moreover, for a commutative ring R, we have shown that R is a Baer-ring if and only if R is a reduced IN -ring. A topological space X is said to be an αU E-space if the closure of any union with cardinal number less than α of clopen subsets is open. Topological properties of αU E-spaces are investigated. Finally, we show that a completely regular Hausdor? space X is an αU E-space if and only if C(X) is an αEGE-ring.     

Banach Spaces with the PC Property

A Banach space X possesses the PC (point of continuity) property if for any w-closed bounded subset A ? X the identity map (A,w)→(A, ∥ ? ∥) has a point of continuity (w is the weak topology in X). We deduce some criteria for Banach spaces to have the PC property and describe (for dual Banach spaces) relationships between spaces possessing the PC property and spaces possessing the RN or the WRN property.

     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

The symplectic Gram-Schmidt theorem and fundamental geometries for A-modules

Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf A is appropriately chosen) shows that symplectic A-morphisms on free A-modules of finite rank, defined on a topological space X, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if (ℰ, φ) is an A-module (with respect to a ℂ-algebra sheaf A without zero divisors) equipped with an orthosymmetric A-morphism, we show, like in the classical situation, that “componentwise” φ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free A-module of finite rank.     

On error bounds for lower semicontinuous functions

We give some sufficient conditions for proper lower semicontinuous functions on metric spaces to have error bounds (with exponents). For a proper convex function f on a normed space X the existence of a local error bound implies that of a global error bound. If in addition X is a Banach space, then error bounds can be characterized by the subdifferential of f. In a reflexive Banach space X, we further obtain several sufficient and necessary conditions for the existence of error bounds in terms of the lower Dini derivative of f. Received: April 27, 2001 / Accepted: November 6, 2001?Published online April 12, 2002     

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

Fraction-dense algebras and spaces

A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fraction-dense f-rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For Archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fraction-dense spaces are defined as those for which C (X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover.     

Why saturated probability spaces are necessary

An atomless probability space (Ω,A,P) is said to have the saturation property for a probability measure μ on a product of Polish spaces X×Y if for every random element f of X whose law is margX(μ), there is a random element g of Y such that the law of (f,g) is μ. (Ω,A,P) is said to be saturated if it has the saturation property for every such μ. We show each of a number of desirable properties holds for every saturated probability space and fails for every non-saturated probability space. These include distributional properties of correspondences, such as convexity, closedness, compactness and preservation of upper semi-continuity, and the existence of pure strategy equilibria in games with many players. We also show that any probability space which has the saturation property for just one “good enough” measure, or which satisfies just one “good enough” instance of the desirable properties, must already be saturated. Our underlying themes are: (1) There are many desirable properties that hold for all saturated probability spaces but fail everywhere else; (2) Any probability space that out-performs the Lebesgue unit interval in almost any way at all is already saturated.     

Representation,interpolation, and reiteration theorems for generalized approximation spaces

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l ^q (ℬ))={f∈X:{E _n (f)}∈l ^q (ℬ)} in which the weighted l ^q -space l ^q (ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces. Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003     

Relative uniform completeness and order-convex representations of archimedean ?-groups and f-rings

Results of Henriksen and Johnson, for archimedean f-rings with identity, and of Aron and Hager, for archimedean ?-groups with unit, relating uniform completeness to order-convexity of a representation in a D(X) (the lattice of almost real continuous functions on the space X) are extended to situations without identity or unit. For an archimedean ?-group, G, we show: if G admits any representation G?D(X) in which G is order-convex, then G is divisible and relatively uniformly complete. A converse to this would seem to require some sort of canonical representation of G, which seems not to exist in the ?-group case. But for a reduced archimedean f-ring, A, there is the Johnson representation A?D(XA), and we show: A is divisible, relatively uniformly complete and square-dominated if and only if A is order-convex in D(XA) and square-root-closed. Also, we expand on the situation with unit, where we have the Yosida representation, G?D(YG): if G is divisible, relatively uniformly complete, and the unit is a near unit, then G is order-convex in D(YG).     

On ideal extensions of partial monounary algebras

In the present paper we introduce the notion of an ideal of a partial monounary algebra. Further, for an ideal (I, f _I ) of a partial monounary algebra (A, f _A ) we define the quotient partial monounary algebra (A, f _A )/(I, f _I ). Let (X, f _X ), (Y, f _Y ) be partial monounary algebras. We describe all partial monounary algebras (P, f _P ) such that (X, f _X ) is an ideal of (P, f _P ) and (P, f _P )/(X, f _X ) is isomorphic to (Y, f _Y ). This work was supported by the Slovak VEGA Grant No. 1/3003/06 and by the Science and Technology Assistance Agency under the contract No. APVT-20-004104.