On [formula omitted]-factorizable topological groups

The fineness index of topological groups and [formula omitted]-factorizable groups

The mixed boundary value problem has been treated recently by several authors [4; 5; 6; 7]. In this note we give an existence proof using subharmonic functions. Consider a 2-dimensional multiply connected domain D with the boundary r, which consists of k closed curves ri, F2, * * *, rk. The boundary curves IX (i =1, 2, * * , k) are assumed to have continuous tangents. A real-valued continuous function f(r) is defined on a part a of the boundary. (The boundary points are denoted by t.) a consists of a finite number of disjoint closed curves or arcs ai. If ai is an arc, then we denote by P', P' its two end points. If(<) | < M. The remaining part of r we denote by ,. Another real valued continuous function g(?) is defined on the closure of ,3. The problem is to determine a function u(z) in D, such that: (a) u(z) is harmonic in D, (b) limz_r u(z) =f(r) on ao and lim supzr I u(z) | < Mat =P', P , (c) limz_r u(z) is continuous for r on , (d) cu/cn=g(v) for r on 3, where the differentiation is along the inward normal n. We shall treat first the problem by assuming g(v) =0.2 Since Neumann's problem can be solved for smooth boundary, the assumption g( )=0 on ( is not a restriction. Define the class 7 of admissible functions v(z) with the properties: (a) v(z) is continuous subharmonic in D, (b) lim supz_r v(z) <f(r) for r on a, (c) lim24 v(z) exists and is continuous for v on A, (d) lim inf Av/An ?0 on (. The lim inf is taken along the normal pointing into the interior of D.

This chapter continues the study of continuous functions from a topological space to the real line space \( \mathbf{R}, \) called the real-valued continuous functions, or, simply, real functions; such functions play a central role in topology and analysis. This chapter also studies uniform convergence of real-valued functions and characterizes normal spaces through separation by real-valued continuous functions.

We prove results on tight asymptotics of probabilities and integrals of the form $P_A (uD)andJ_u (D) = \int\limits_D {f(x)\exp \{ - u^2 F(x)\} dP_A (ux)} $ , where P A is a Gaussian measure in an infinite-dimensional Banach space B, D = {x ? B: Q(x) ? 0} is a Borel set in B, Q and F are continuous functions which are smooth in neighborhoods of minimum points of the rate function, f is a continuous real-valued function, and u?? is a large parameter.

A remark on decomposition of continuous functions

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

A now standard technique for treating a class of Baire set problems, originating with Halmos [2; 51.D], is to construct a metric space that, in a way, "fixes" a given Baire set. Formally this means, given a Baire set B in a space X, find a metric space M and a map g of X onto M with the property that B = g l gB . In phrasing the problem for topological groups it is natural to require that M be a group and g be a homomorphism. Specifically, what is desired is a subgroup H that "fixes" B, i.e., such that B = BH. This problem is solved in Halmos for locally compact a-compact groups in a theorem [2; 64.G] that has led to interesting consequences elsewhere [4, 1]. As will be seen, Halmos' theorem is also true for totally bounded groups. An example of Sirota's of a countable topological group with no non-trivial metrizable homomorphic images indicates that, in this direction at least, the barrier of local compactness cannot be lifted without giving up metrizability in M. If this is done, however, replacing the requirement that M be metrizable, i.e., that H have a countable base in G, with the less stringent requirement that H be a G~-set in G, an analogue of Halmos' theorem is true for Lindelff groups. In view of these remarks it is convenient to distinguish between two distinct Baire set fixing problems. A closed invariant subgroup H of a group G is said to be a fixiru3 subgroup (M-fixing subgroup) provided H is a G~-set (H has a countable base of neighborhoods in G). G satisfies the Baire set fixing property (M-Baire set fixing property) provided, for each Baire set in G, there is a fixing subgroup (M-fixing subgroup) such that B = BH. The Baire sets are taken to be the elements of the smallest a-algebra containing the zero sets of continuous real-valued functions on G. For technical convenience in describing proofs, a G~-set containing the identity is called an admissible set.

All Hartman–Mycielski groups are minimally almost periodic

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions<br /> on $L$.<br /> We consider the set $$mathcal{R}_{infty}L = {varphi in mathcal{R} L : uparrow varphi( dfrac{-1}{n}, dfrac{1}{n})<br /> mbox{ is a compact frame for any $n in mathbb{N}$}}.$$<br /> Suppose that $C_{infty} (X)$ is the family of all functions $f in C(X)$ for which the<br /> set ${x in X: |f(x)|geq dfrac{1}{n} }$<br /> is compact, for every $n in mathbb{N}$.<br /> Kohls has shown that $C_{infty} (X)$ is precisely the intersection<br /> of all the free maximal ideals of $C^{*}(X)$.<br /> The aim of this paper is to<br /> extend this result to<br /> the real continuous functions on a<br /> frame and hence we show that $mathcal{R}_{infty}L$ is precisely the intersection<br /> of all the free maximal ideals of $mathcal R^{*}L$.<br /> This result is used to characterize the maximal ideals in $mathcal{R}_{infty}L$.

On bM-ω-balancedness and [formula omitted]-factorizability of para(semi)topological groups

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

A fading-memory system is defined as a system whose response map has unique asymptotic properties over some set of inputs. It is shown that any discrete-time fading-memory system can be uniformly approximated arbitrarily closely over a compact set of input sequences by uniformly approximating either its external or internal representation sufficiently closely. In other words, the problem of uniformly approximating a fading-memory system reduces to the problem of uniformly approximating continuous real-valued functions on compact sets. The perceptron is shown to realize a set of continuous real-valued functions that is uniformly dense on compacta in the set of all continuous functions. Using the perceptron to uniformly approximate the external and internal representations of a fading-memory system results, respectively, in simple nonlinear finite-memory and infinite-memory system models.

Dimension and superposition of bounded continuous functions on locally compact, separable metric spaces

Topological pressure and large deviations for multivariate potentials

Introduction. For a metric space (X, d) the following three levels of uc-ness (continuity of some set of functions is uniform) coincide: (a) any real-valued continuous and bounded function is uniformly continuous, i.e. C*(X ,R) = U*(X,R); (b) any real-valued continuous function is uniformly continuous, i.e. C ( X , R ) = U(X,R); (c) any continuous function from X to any (metrizable) uniform space is uniformly continuous, i.e. C ( X , Y ) = = U(X, Y ) for any uniform space Y. The equivalence is due essentially to the following sequential characterization of uniform continuity in the metric case. A function f : (X, d) --+ (II, d') is uniformly continuous iff for each pair ofsequences (xn), ( yn ) o f X iflim d( xm y, ) = O then fim d' ( f ( z , ), f ( yn ) ) = O. The uniform version of the previous sequential characterization of uniform continuity has been used to define sequential uniform spaces. The category of sequential uniform spaces introduced by Hu~ek [4] is a wide class of uniform spaces including metric spaces, closed under sums and quotients. A sequentially uniform space is a remarkable example of uniform space in which if any real-valued continuous and bounded function is uniformly continuous then any continuity is uniform. We will obtain this result proving that a sequential uniformity is the largest member of its proximity class. Further, after generalizing in a natural way the notion of pseudo-Cauchy sequence , we will show that a normal sequential uniformity is fine (any continuity is uniform) iff any pseudo-Cauchy sequence with distinct points has a cluster point.