Axiom Of Countability Research Articles

A geometric characterization of non-atomic measure spaces

Professors J. B. Conway and J. Sztics have kindly pointed out to the author that there is a gap in the proof of Theorem I in the paper [2]. The oversight occurs in the first sentence on p. 58 of [2], where it is asserted that "... the closure is mid-point convex". Due to the fact that the ball U(L°~(/~)) (all unexplained notation is that of [2]) need not satisfy any countability axioms with respect to the weak* topology, the above assertion only becomes convincing under some additional hypothesis such as "the space L~(#) is separable". The purpose of the present note is to indicate how the technique introduced in [2] can be further exploited to yield a valid proof of Theorem I as originally formulated. A persual of the argument given in [2] shows that what is needed for completeness is a proof of the following Assertion. Let (S, Z, #) be a finite non-atomic measure space and let 1 < 2 < 1. Then there is a sequence {f,} C ext{U(L~(~u))) such that

Remarks on the reduction theory of von Neumann algebras

In [7, Problem 4, p. 3.55], S. Sakai proposed the following question: If any component of a measurable field -yi-*M('y) of von Neumann algebras over (r, bt) is isomorphic to some fixed von Neumann algebra oZo with separable predual; where r means a locally compact Hausdorff space and IA a positive Radon measure on r, is it true that the direct integral f M< l(,y)d,u('y) is isomorphic to the tensor product a?0 Mo of the associated diagonal algebra'(=L(r, I) and So? If r satisfies the second countability axiom, then one can easily settle the above problem affirmatively as an application of [1, Proposition 4, p. 187]. However, in the case that the countability assumption of r is dropped, J. Dixmier proposed the similar problem in [1, p. 175]. Namely, roughly speaking, if zyl-yj(,y) is a measurable operator field over (r, IA) for each iFI and there exists a family of bounded operators {xi}irr on a Hilbert space and an isometric operator u(,y) for each y such that u('y)xiu('y)--=yi(y) for every yr and iCI, does there exist an isometry which transforms 1 Xxi to f ED yi(,y)du('y) for every

Almost Continuity of Mappings

Let E be a metric Baire space and f a real valued function on E. Then the set of points of almost continuity in E is dense (everywhere) in E.Our purpose is to set this result in its most natural context, relax some very restricted hypotheses, and to supply a direct proof. More precisely, we shall prove that the metrizability of E in Theorem H may be removed, and that the range space may be generalized from the (Euclidean) space of real numbers to any topological space satisfying the second axiom of countability [2].

A Remark on the Continuity of the Dual Process

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

Homotopy types of ANR’s

PROOF. Suppose not. Let {A, } be an uncountable collection of different types. We may regard each AO as a subset of the Hilbert cube I@. For each : let c be chosen so that AO is a retract of its en-neighborhood UB in Iw. Let r,: UO-*A, be a retraction map. Since {AO is uncountable we may assume (by choosing a subcollection) that there is a positive e so that E> e for each ,B. Note that if f and g are any two maps of a space X into an AO so that the distance in 1X between f(x) and g(x) is less than E, for all x in X, thenf and g are homotopic. In fact F(x, t) -r#((1 -t)f(x) +tg(x)) is the desired homotopy, where we use the linear structure in I@'. For each ,B there is a 8> 0 so that r, restricted to the 8-neighborhood of AO moves no point as far as e/2. Again we may assume that there is a positive a so that 8! ?8 for each S. The hyperspace of all closed subsets of IX with the Hausdorff metric is separable (e.g. if D is any countable dense set in I@1, the collection of finite subsets of D is countable and dense in the hyperspace). Since every uncountable set, in a space satisfying the Second Axiom of Countability, contains a limit point, we see that for some : and y, AO is in the 8-neighborhood of A, and vice versa. Then rj IA and r,J A, are homotopy inverses of each other, since their compositions in either direction move points a distance less than E, hence, are homotopic to the corresponding identity maps. This shows A3 and A. have the same homotopy type, a contradiction.

On isomorphisms with small bound

If X and Y are locally compact Hausdorff spaces, and if we denote by Co(X) and CO( Y) the Banach spaces of continuous, complexvalued functions vanishing at infinity on X and Y respectively, then, according to the Banach-Stone Theorem, the existence of an isometry between the function spaces Co(X) and Co(Y) implies that X and Y are homeomorphic. In [1 ] it was shown that if we assume, in addition, that X and Y satisfy the first axiom of countability, then this theorem can be generalized by considering isomorphisms with small norm instead of isometries. The object of this paper is to show that the conclusions of [1] remain valid without the assumption of first countability.

On Extensions of Locally Compact Groups

Let Q be a locally compact group, and let K be a Q-module, that is, a locally compact Abelian group on which Q operates continuously as a group of automorphisms. An extension of Q by K is a locally compact group, G, together with a homomorphism, p, of G onto Q, and a topological isomorphism, i, of K onto the kernel of p, such that p induces a topological isomorphism of G/i (K) onto Q in such a way that the natural action of Q G/i (K) on K i(K) by inner automorphisms is the given action of Q on K. (See [8] for an extensive bibliography). Mackey [6] showed that if K and Q are separable, in the sense of satisfying the second axiom of countability, then to each extension, G, of Q by K there corresponds a (non-unique) normalized Borel 2-cocycle on Q with values in K, that is, a Borel function c from Q X Q into K satisfying the cocycle identity

Closed set countability axioms

Closed set countability axioms

Characterizations of Developable Topological Spaces

The class of developable topological spaces, which includes the metrizable spaces, has been fundamentally involved in investigations in point set topology. One example is the remarkable edifice of theorems relating to these spaces constructed by R. L. Moore (13). Another is the role played by the developable property in several metrization theorems, including Alexandroff and Urysohn's original solution of the general metrization problem (1).This paper presents an anslysis of the concept of developable space in terms of certain more extensive classes of spaces satisfying the first axiom of countability : spaces with a base of countable order and those having what is here called a θ-base. The analysis is given in the characterizations of Theorems 3 and 4 below.

On open mappings and certain spaces satisfying the first countability axiom

On open mappings and certain spaces satisfying the first countability axiom

A spinor formulation of electromagnetic theory in general relativity

The spinor formulation of the Maxwell-Lorentz theory that was developed earlier bySachs andSchwebel (2–3) is extended to a generally covariant formalism. The underlying group for electrodynamics is thereby generalized from the proper inhomogeneous Lorentz groupLp to a special topological groupT. It follows from the physical assumptions of this theory that the groupT must belocally compact, connected, and satisfy thesecond axiom of countability. The invariants and conservation laws of the generalized theory are derived and a correspondence respondence of some of them with those of the conventional theory is demonstrated. The remaining invariants and conservation laws have no counterpart within the usual formalism. Extra terms that appear are due to the spin-affine connection field and the invariants which do not conserve reflection symmetry in space and time. Within this generalized formalism, the uniqueness of earlier derived results of a generalized Lorentz covariant theory of electromagnetism (13) is demonstrated. The Lagrangian density for the present formulation is derived and the role that is played by a fundamental length, which is inherent in the theory, is discussed.

The existence of complete Riemannian metrics

The purpose of the present note is to prove the following results. Let M be a connected differentiable manifold which satisfies the second axiom of countability. Then (i) M admits a complete Riemannian metric; (ii) If every Riemannian metric on M is complete, M must be compact. In fact, somewhat stronger results will be given as Theorems 1 and 2 below. Let M be a connected differentiable manifold. It is known that if M satisfies the second axiom of countability, then M admits a Riemannian metric. Conversely, it can be shown that the existence of a Riemannian metric on M implies that M satisfies the countability axiom. For any Riemannian metric g on M, we can define a natural metric d on M by setting the distance d(x, y) between two points x and y to be the infinimum of the lengths of all piecewise differentiable curves joining x and y. The Riemannian metric g is complete if the metric space M with d is complete. It is known that this is the case if and only if every bounded subset of M (with respect to d) is relatively compact. We shall say that a Riemannian metric g is bounded if M is bounded with respect to the metric d. We shall prove

On the first countability axiom for locally compact Hausdorff spaces

On the first countability axiom for locally compact Hausdorff spaces

Moore spaces and uniform spaces

Although arising from different motivations Moore spaces2 and uniform spaces [7] are generalizations of metric spaces which have considerable similarity. This can be seen from the following characterizations by means of open coverings: Let S be a regular Hausdorff space. If there exists a family { G} of open coverings of S such that (a) { G } is countable and3 (b) if U is an open set and p G U, there is an element G of { G} such that Gp*C U, then S is a Moore space and conversely [6]. If there exists a family { G} of open coverings of S such that (a') if G, and G2 are elements of { G} there is an element G3 of { G } which is a star refinement of both G, and G2 and (b) if U is an open set and p GU, there is an element G of { G such that GP*C U, then S is a uniform space4 and conversely [5]. Of course if a { G} exists so that all three conditions (a), (a') and (b) hold true simultaneously, S is metric.' Except for the obvious fact that the Hausdorff first countability axiom must hold true for Moore spaces while it need not do so for uniform spaces, it is by no means obvious that they differ otherwise. L. F. McAuley has given an example of a normal semi-metric Hausdorff space S which is not a Moore space [3]. Being normal, S is both regular and uniform and being semi-metric, the first countability axiom holds true for S. I give below an example of a locally connected, connected, complete Moore space6 containing a point p at which it is not completely regular. Hence it is not uniform [2 or 5 ]. This space is obtained by piecing together along their boundaries adjacent terms of a simple sequence

Some remarks on a paper of Aronszajn and Panitchpakdi

In the paper of the title [1], a number of problems are posed. Ne­ gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self­ contained. 1. A Hausdorff space X is said to satisfy (Qm), where m is an in finite cardinal, if, whenever U and V are disjoint open subsets of X such that each is a union of the closures of less than m open subsets of X, then U and V have disjoint closures. In particular, a normal (Hausdorff) space X satisfies (Q*,) if and only if disjoint open F.-SUbsets of X have disjoint closures. (For, an open set that is the union of less than ~, closed sets is a fortiori an F.. Conversely if U is the union of countably many closed subsets Fo> then since X is normal, for each n there is an open set Un containing Fn whose closure is contained in U. Thus U is the union of the closures of the open sets Un.) In Prob­ lem 3 of [1], it is asked if every compact (HaUSdorff) space satisfying (Qm) for some m>~o is necessarily totally disconnected, and it is re­ marked that this is the case if the first axiom of countability is also as­ sumed. If X is a completely regular space, let C(X) denote the ring of all continuous real-valued functions on X, and let Z(f)= {x EX: f(x)=O}, let P(f)={xEX:f(x»O}, and let N(f)=P(-f). As usual, let (IX denote the Stone-Cech compactification of X. If every finitely generated ideal of C(X) is a principal ideal, then X is called an F-space. The fol­ lowing are equivalent. ( i) X is an F-space. ( ii) If f E C(X), then P(f) and NU) are completely separated [2, Theorem 2.3]. (iii) If f E C(X), then every bounded g E C(X-ZU» has an ex­ tension g E C(X) [2, Theorem 2.6]. A good supply of compact F-spaces is provided by the fact that if X is locally compact and ,,-compact, then {lX-X is an F-space [2, Theo­ rem 2.7].

On rings of bounded continuous functions with values in a division ring

Introduction. Let C*(X, A) denote the ring of bounded continuous functions on a (Hausdorff) topological space2 X with values in a topological division ring A. If, for every maximal (two-sided) ideal M of C*(X, A), we have C*(X, A)/M is isomorphic with A, we say that Stone's theorem holds for C*(X, A). It is well known [9; 6] that Stone's theorem holds for C*(X, A) if A is locally compact and connected, or a finite field. In giving a partial answer to a question of Kaplansky [7], Goldhaber and Wolk showed in [5] that, with restriction on X, and if A is of type V and satisfies the first axiom of countability, then a necessary condition that Stone's theorem holds for C*(X, A) is that A be locally compact. They ask if this condition is also sufficient. Below, we answer this question in the affirmative. In the course of their proof of the necessity of this condition, they construct a class of maximal ideals of C*(X, A). (See ?3). They also ask if every maximal ideal of C*(X, A) is in this class. We show that even if both X and A are the reals, the answer to this question is no.

On Continuity and Openness of Homomorphisms in Topological Groups

In the present paper we wish to give certain necessary and sufficient conditions that a homomorphism h between two topological groups X and Y be continuous, and similar conditions that it be open, i.e., map open sets into open sets. The present results may be considered as extensions to less restricted groups of theorems known to hold in Banach spaces or in spaces of Type (F) or in separable groups. The guiding principle has been that of transferring as much as possible the restrictions in these earlier theorems from the groups to the homomorphism. In Part I, after extending Banach's theorem on second category Baire subgroups to show (Theorem 1) that the identity element always lies in the of A_'A whenever A contains a second category Baire set, it is proved in Theorem 2 that h is continuous and X second category if and only if h-'(V) is either null or contains a second category Baire set whenever V is open in Y; this is easily dualized to show that h is open and Y second category if and only if h(U) contains a second category Baire set for each non-null open U in X. From these there follow theorems given by Banach [2, 3], Bourbaki [6], Freudenthal [9], and McShane [15]. There are also extensions to linear topological spaces of certain results of Banach [2] and of Mazur and Orlicz [14] for Banach spaces and spaces of Type (F). In Part II our methods, based on a proof of Banach's, require that X and Y satisfy the first countability axiom. The essential result here (Theorem 15) is that if X is right complete, i.e., complete with respect to at least one of its rightinvariant metrics, then h is open and has a kernel if and only if it has a and maps non-null open sets into somewhere dense sets. The dual of this also holds (Corollary 16.1); if Y is right complete and is a Hausdorff group then h is continuous if and only if it has a and h-' maps each open set in Y into either the null set or a somewhere dense set. These are extensions of Banach's interior mapping and closed graph theorems in Type (F) spaces [2]. A theorem of Dunford's [8] also follows with weaker hypotheses. In conclusion there are indications of some extensions of the present results. By a topological space (TS) we shall understand as usual a space X with a distinguished class of subsets, called open sets, which is under arbitrary

On the Structure of Complete Local Rings

The concept of a local ring was introduced by Krull [2], who defined it as a Noetherian ringR(we say that a commutative ringRis Noetherian if every ideal inRhas a finite basis and ifRcontains the identity) which has only one maximal idealm. If the powers of m are defined as a system of neighbourhoods of zero, thenRbecomes a topological ring satisfying the first axiom of countability, And the notion was studied recentlybyC. Chevalley and I. S. Cohen. Cohen [1] proved the structure theorem for complete rings besides other properties of local rings.

Compact sets of functions and function rings

A widely used theorem of analysis asserts that a uniformly bounded, equicontinuous family of functions has a compact closure in the space of continuous functions. This lemma, variously attributed to Arzela, Escoli, Montel, Vitali, and so on, is of importance in the theory of integral equations, conformal mapping, calculus of variations, and so on. In recent years the lemma has been generalized by S. B. Myers [1 ].I A part of his results may be formulated as follows; If a topological space X is either (a) locally compact, (b) satisfies the first axiom of countability, and if Y is a metric space, then a family F of continuous functions from X to Y is compact (in a suitable topology) if and only if (1) F(x) = UfEFf(x) is compact for all xCX, (2) F is closed, (3) F is equicontinuous. The main purpose of ?1 of this paper is to characterize compact sets of functions when Y is any regular topological space. The problem is therefore to find a condition to replace equicontinuity, which no longer makes sense. We obtain such a characterization which holds for an easily described class of spaces X which includes both locally compact and first countable. In ?2 these results are applied to obtain a sort of duality theorem for the ring of real-valued continuous functions, R(X), on a space X. Namely, it is shown that, under quite general conditions, the space X is homeomorphic with the space H(X) of continuous homomorphisms from the ring R(X) onto the real numbers, where R(X) and H(X) are given the compact-open topology.

Two-ended topological groups

Introduction. Let G be a locally compact, connected topological group (satisfying the second countability axiom). Let G* be a compact space which contains a dense subset G' homeomorphic to the space G and is such that G*-G' is totally disconnected. Then, Freudenthal has proved [1, Satz 1X, p. 277]1 that the set G*-G' consists of at most two distinct points. Actually, Freudenthal's theorem even for topological groups is more general than here stated, and this theorem is an application to group spaces of a wider theory of ends of topological spaces. However, we shall quote only so much of Freudenthal's results as are necessary to this paper. It will be convenient to regard G' as identical with G so that G is topologically imbedded in G*. We shall call a locally compact, connected group G two-ended if a G* exists such that G* G consists of two distinct points. The simplest example of such a group is the additive group of reals. Other examples are afforded by the direct product of this group and any compact connected topological group; it is likely that these are the only examples. The principal objective of this note is the following theorem.