C-16 - Corson Compacta

b-7 - Cleavable (Splittable) Spaces

We assume a basic knowledge of general topology and integration theory (as found, for example, in [91], [104], [130], and [253]). The purpose of this chapter is to present some special results which are not necessarily found in general references. In section 4 we prove an interpolation theorem and investigate when compact Hausdorff spaces can be mapped continously onto the closed unit interval [0, 1]. A brief development of dispersed spaces and their relationship to spaces of ordinal numbers is given in section 5. Section 6 is devoted to a study of the Cantor set and section 7 is concerned with extremally disconnected compact Hausdorff spaces and their role as projectives (in the category of compact Hausdorff spaces and continuous maps). In section eight we briefly develop the theory of regular Borel measures and prove representation theorems for C(T,ℝ)* (and C(T,ℂ)*).

Some aspects of dimension theory for topological groups

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

A basic theorem asserts that a continuous function on a compact metric space with values in another metric space is uniformly continuous. The usual proofs based on a contradiction argument involving sequences or on the covering property of compact sets are quite sophisticated for students taking a first course on real analysis. We present a direct proof only using results that are established anyway in such an introductory course. Let f ∶ X → Y be a continuous function from the compact metric space (X, dX) into the metric space (Y , dY ). The function F ∶ X ×X → R given by F (x, y) ∶= dY (

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

Comparing Cartesian closed categories of (core) compactly generated spaces

The topological fundamental group and free topological groups

Function-space compactifications of function spaces

We give a complete description of the topological spaces X such that the free abelian topological group A(X) embeds into the free abelian topological group A(I) on the closed unit interval. In particular, the free abelian topological group A(X) on any finite-dimensional compact metrizable space X embeds into A(I). To obtain our description, we study similar embeddings of the free locally convex spaces and continuous surjections between the spaces of continuous functions with the pointwise topology. Proofs are based on the classical Kolmogorov's Superposition Theorem.

Chapter 52 - The interplay between compact spaces and the Banach spaces of their 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.

d-20 - H-Closed Spaces

Let X be a compact Hausdorff space and C(X) the Banach space of all real-valued continuous functions on X under the sup norm. If H is a linear subspace of C(X) which separates the points of X and contains the constant functions, then it is well known that there exists a smallest closed subset of X, called the Silov boundary of X relative to H and denoted by OHX, with the property that each hCH attains its maximum value on afHX. A point xeX is called an H-extremal point if the only positive linear functional u on C(X) such that u(h) = h(x) for all hCH is the evaluation functional 4. where O.,(f) =f(x) for all fE C(X). The set of H-extremal points of X is called the Choquet boundary of X relative to H and will be denoted by VHX. H. Bauer has shown [1, 2.1 ] that the Choquet boundary of X relative to H is nonempty and the Silov boundary of X relative to H is equal to the closure of the Choquet boundary. Bauer has also introduced (see [1 ]) an abstract Dirichlet problem for the above setting. If S:DHX is closed and xCX, then MS denotes the set of positive linear functionals u on C(S) such that u(h I S) =h(x) for all hCH. MS is always nonempty. The measures in MX'HX are called the H-harmonic measures belonging to x. A function f in C(X) is said to be H-harmonic if for every xEX and every u M we have u(f) =f(x). The set of H-harmonic functions is denoted by H. The Dirichlet problem is said to be solvable for X relative to H if 1^ j aHX = c(aHX). The Dirichlet problem is solvable if and only if to each x in X belongs exactly one H-harmonic measure (see [1, Satz 9]). Let X1, X2 be compact Hausdorff spaces and H1, H2 separating linear subspaces of C(X1), C(X2) respectively, which contain the constant functions. Equipped with the usual product topology, the cartesian product X1 X X2 is a compact Hausdorff space. In this paper we show that a subfamily H1+H2 of C(XIXX2) can be defined, in a natural way, for which VH1+H2Xl X X2 exists and VH1+H2Xl X X2 = VH1X1X VH2X2. We then give two theorems on the relation of the

All Hartman–Mycielski groups are minimally almost periodic