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In order to analyse generic or typical properties of dynamical systems we consider the space of all C1-vector fields on a fixed differentiable manifold M. In the C1-metric, assuming M is compact, is a complete metric space and a generic subset is an open dense subset or an intersection of a countable collection of such open dense subsets of . Some generic properties (i.e. specifying generic subsets) in are described. For instance, generic dynamic systems have isolated critical points and periodic orbits each of which is hyperbolic. If M is a symplectic manifold we can introduce the space of all Hamiltonian systems and study corresponding generic properties.

A method of “maximum variance nondimensional scaling” is described and tested that transforms similarity measures into distances that meet just three conditions: (C1) they exactly satisfy the metric axioms, (C2) they are, as nearly as possible, monotonically related to the similarity measures, (C3) they have maximum variance possible under the two preceding conditions. By achieving an appropriate balance between the last two conditions, one can determine the true underlying distances and the form of the unknown monotone function relating the similarity measures to those distances without assuming that the underlying space has any particular Euclidean, Minkowskian, or even dimensional strucutre. The method appears to have potential applications, e.g., to studies of stimulus generalization and the structure and processing of semantic information.

A space Y has the countable chain condition (CCC) (discrete countable chain condition (DCCC)) provided each (discrete) collection of mutually exclusive open sets in Y is countable. In [16], the author showed the DCCC, the CCC and separability to be equivalent conditions in completable Moore spaces. However, in [18], Rudin gave an example of a non-separable Moore space with the CCC. And in [15], under the assumption of the Continuum Hypothesis, the author gave an example of a Moore space with the DCCC but not the CCC.In this paper, the author gives an example of a Moore space S with the DCCC but not the CCC which requires no set-theoretic assumptions other than the Axiom of Choice. The construction of this space is of a general nature (to each first-countable T3 space are associated two Moore spaces), which the author believes will be a useful technique in the search for other counterexamples. The space S is also shown to be a pseudonormal Moore space which does not have the three link property. In addition, the author gives characterizations of chain conditions in Moore spaces and relates these conditions to Moore closure and pseudocompactness.

In this paper, we give necessary and sufficient conditions for embedding a given metric space in Euclidean space. We shall introduce the notions of flatness and dimension for metric spaces and prove that a metric space can be embedded in Euclidean n-space if and only if the metric space is flat and of dimension less than or equal to n.

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product. Because I have found a countable box product of ordinals useful in the past [1], this class of spaces particularly interests me. The purpose of this paper is to tell what I know about which of these spaces is paracompact or normal. In [2] I prove that the continuum hypothesis implies the box product of countably many a-compact, locally compact, metric spaces is paracompact. I prove here that the continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact (Theorem 1) and the box product of another class of ordinals is normal (Theorem 2). The proof of Theorems 1 and 2 is a quite messy join of the techniques of [1] and [2] which raises some doubt in my mind as to whether these theorems are worth proving. Because I care, because I think these spaces are set theoretically interesting and topologically useful, because I think these theorems are best possible, the theorems are worth the mess to me. A. If {XXAOA)\ is a family of topological spaces, a box in HIAeA XA is a set TheA UA where each UA is open in XA. The box product of {XAIA A iS HAA XA topologized by using the set of all boxes in it as a basis. Throughout the paper the following notation is used. An ordinal a is the set of all ordinals less than a and a is topologized by the interval topology. The statement that a is a cardinal means that a is an ordinal and no smaller ordinal has the same cardinality as a. The notation IAnA BAA is used to mean the ordinary Cartesian product of the f,3's and never the cardinal or ordinal arithmetic product. Similarly a#9 means the set of all functions from ,B into a rather than an arithmetic operation. If a is an ordinal, let cf(a) denote the cofinality of a; that is cf(a) is the smallest ordinal 8 such that there is a subset A of a, order isomorphic with 8, such that /3 < a implies there is a y El A with /3 < y. Observe that a is a a-compact ordinal if and only if a is compact or cf(a) =w. Received by the editors December 10, 1971 and, in revised form, October 1, 1972. AMS (MOS) subject classiflcations (1970). Primary 54B10, 54A25, 54D15, 54D20, 54D30, 02K25.

It is proved that the arbitrary product of arcwise connected spaces is arcwise connected. Introduction. By an arc we mean a Hausdorff continuum with at most 2 noncut points, called the end points of the arc. A space S is said to be arcwise connected if whenever x, y E S, then x and y are the end points of some arc in S. It is well known (see, for instance, [4, Theorems 28.8 and 28.13]) that a nondegenerate metric continuum A is an arc if and only if A is homeomorphic to [0, 1]. Since a metrizable product of arcs is a compact, connected and locally connected metric space, it follows [4, Theorem 31.2] that a metrizable product of arcs is arcwise connected. However, examples have been constructed by S. Mardesic [2] and [3] and J. L. Cornette and B. Lehman [1] of locally connected Hausdorff continua which are not arcwise connected. Thus the above argument will not suffice for a nonmetrizable product of arcs, even if each factor space is metrizable. In this paper we show that the arbitrary product of arcwise connected spaces is arcwise connected. LEMMA Let {X,: a E W} be a collection of nondegenerate arcs, and let X denote the product space of this collection. If the end points of Xa are a. and ba, then there is an arc in Xfromf to g ii-heref is that pointfor which f(c)=a. and g is that point for which g(oc)=ba. PROOF. Let ? be a well-ordering of s', and let "1" denote the first element of a?, and "oc+ 1" the successor of a. in a. For each c E a., define the "edge" A. of X and pointsfa and g. of X as follows: Aa = {h e X:h(/3) = b#,5 c., = apl > a; =ap , > a. Received by the editors May 4, 1973. AMS (MOS) subject classifications (1970). Primary 54B10, 54F05, 54F20.

The purpose of this paper is to study a class of metric spaces over a field of characteristic > 2 and the groups of their motions.

In the paper “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl. 9 (1964), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X.

A continuum (i.e., a compact connected nondegenerate metric space) M M is said to be λ \lambda connected if any two of its points can be joined by a hereditarily decomposable continuum in M M . Here we prove that a plane continuum is λ \lambda connected if and only if it cannot be mapped continuously onto Knaster’s chainable indecomposable continuum with one endpoint. Recent results involving aposyndesis and decompositions to a simple closed curve are extended to λ \lambda connected continua.

The purpose of this paper is to represent a class of metric spaces on the elliptic and hyperbolic inversive planes by inversions and their transformations.

The purpose of this note is to announce the major ideas and results developed in [R]x. The proofs of these results will appear in a series of three papers [R]2, [R]3, and [RR], the latter including categorical topics that will be omitted here. The subject matter is the covering and function theoretic properties of uniform spaces, a subject initiated by John Isbell in the 1950's. (See [GI] and [I].) Our work represents a continuation and extension of the current work of Anthony Hager ([H]l5 [H]2) and Z. Frolik; and overlaps somewhat with recent work of Z. Frolik ([Fr]l5 [Fr]2). The author wishes to emphasize that his work substantiates the existence of a theory of uniform structures which is not primarily interested in topological applications. Therefore, the viewpoint adopted here is one of intrinsic interest per se in uniform properties. A uniform space is denoted by uX, where u is a family of covers on the set X constituting a uniformity. uX is fine if u is the largest uniformity on X with the same uniform topology. A subfine space is a subspace of a fine space. uX is locally fine if each cover of the form {AanCp} eu, where {Aa} e w, and {Cp} e u for each a. uX is M-fine (sub-M-fine) if each uniformly continuous function (map) to a metric (complete metric) space remains a map relative to the fine uniformity on M (the uniformity with the basis of open covers of M). uX is hereditarily M-fine if each subspace is M-fine. The basic source on locally fine and subfine spaces is [I], while the development of separable M-fine and separable hereditarily M-fine spaces (those with a basis of countable covers) originates in [Hh and [H]2. One easily sees that each fine space is M-fine and that each M-fine space is sub-M-fine. Example C of [GI] is a hereditarily M-fine space which is not locally fine. [I] shows that each locally fine space is sub-M-fine and that each subfine space is locally fine; the converse of the latter is an unsolved problem. From [I] we also know that each separable sub-M-fine space is subfine.

The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences mod k \bmod \;k . A space X has a refining sequence mod k \bmod \;k if there exists a sequence { G n | n ∈ N } \{ {\mathcal {G}_n}|n \in N\} of open covers for X such that ∩ n = 1 ∞ St ( C , G n ) = P C 1 \cap _{n = 1}^\infty {\text {St}}(C,{\mathcal {G}_n}) = P_C^1 is compact for each compact subset C of X and { St ( C , G n ) − | n ∈ N } {\text {\{ St}}{(C,{\mathcal {G}_n})^ - }|n \in N\} is a neighborhood base for P C 1 P_C^1 . If P C 1 = C P_C^1 = C for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if P C 1 = C P_C^1 = C for all such sets then X is developable. Thus the concept of a refining sequence mod k \bmod \;k is natural and it is helpful in understanding paracompact p-spaces.

A Lagrangian-based metric theory of gravity is developed with three adjustable constants and two tensor fields, one of which is a nondynamic 'flat space metric' eta. With a suitable cosmological model and a particular choice of the constants, the 'post-Newtonian limit' of the theory agrees, in the current epoch, with that of general relativity theory (GRT); consequently the theory is consistent with current gravitation experiments. Because of the role of eta, the gravitational 'constant' G is time-dependent and gravitational waves travel null geodesics of eta rather than the physical metric g. Gravitational waves possess six degrees of freedom. The general exact static spherically-symmetric solution is a four-parameter family. Future experimental tests of the theory are discussed.

The Rabins cut-point theorem gives some conditions on acceptance of a probabilistic langnage by a finite state automaton. In the present, paper we will generalize Rabin's results by studying the possibility of reducing a coutext-free or right-linear grammar with infinite non-terminal symbols to an equivalent grammar with a finite number of non-terminal symbols. We will show that reducibility is related to some topological properties of non-terminal symbols, which we will consider as a metric space.

The analogy between dist (a, b)+dist (b, c)≥dist (a, c) and hom (A, B) ⊗ hom (B, C)→hom (A, C) is rigorously developed to display many general results about metric spaces as consequences of a «generalized pure logic» whose «truth-values» are taken in an arbitrary closed category.

Let X be a compact connected metric space and 2X(C(X)) denote the hyperspace of closed subsets (subcontinua) of X. In this paper the hyperspaces are investigated with respect to point-wise connectivity properties. Let MeC(X). Then 2X is locally connected (connected im kleinen) at M if and only if for each open set U containing M there is a connected open set V such that M

An intrinsic characterization of the range onto which each open map from each metric space becomes compact-covering is given.

The parallelizability of the dynamical systems is used in order to characterize those connected, locally compact, metric spaces, which have the following property: there exists at least a metric, such that the unit's component of the corresponding group of isometries (with the compact-open topology) is not compact; they are exactly those metric spaces, which can be presented as R×Y (:R denotes the space or the group of the reals).

The paper provides necessary and sufficient conditions for a weakly zero-dimensional metrizable space to be strongly paracompact, i.e., to have the star-finite property. The characterizations use special basis properties of uniformities which induce the topology of X, and yield further characteristics of the class of all metric spaces with ind X = 0 X = 0 and Ind X > 0 X > 0 .

Let T : M → M T:M \to M be a mapping of a metric space ( M , d ) (M,d) into itself. A mapping T T will be called a quasi-contraction iff d ( T x , T y ) ⩽ q max { d ( x , y ) ; d ( x , T x ) ; d ( y , T y ) ; d ( x , T y ) ; d ( y , T x ) } d(Tx,Ty) \leqslant q\max \{ d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\} for some q > 1 q > 1 and all x , y ∈ M x,y \in M . In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach’s contraction principle. Multi-valued quasi-contractions are also discussed.

In this note we generalize a theorem of Erdos and Szekeres, which states that every sequence of real numbers of length n 2 + 1 has a monotone subsequence of length n + 1, for points in certain metric spaces ( R k , d ), where d is a Minkowski metric. Three theorems are proved concerning preassigned numbers of points which must lie on the same geodesic of the space, the last of which characterizes the class of Minkowski spaces under discussion.

An approximate metric is found which represents a sphere of matter embedded in a background of dust. The use of this metric in conjunction with the Friedmann equations gives values of Λ for the three possible values ofk as Λ≃+6×10−36 (k=+1), Λ≃+3×10−35 (k=0), Λ≈+10−36 (k=−1). These values depend on data regarding clusters of galaxies, and are probably accurate to within an order of magnitude given the correctness of the assumptions on which their derivation rests.

It is shown that a Hausdorff continuum S S is the continuous image of an arc (respectively arcwise connected) if and only if each cyclic element of S S is the continuous image of an arc (respectively, arcwise connected). Also, there is given an analogue to the metric space cyclic chain approximation theorem of G. T. Whyburn which applies to locally connected Hausdorff continua.

The work presented in this paper was initially motivated by the following question of A. Wilansky: “Is there a smallest FK-space E in which is bounded?” Here FK-space means a complete linear metric space of real or complex sequences x = (xi) upon which the coordinate functional x → xt are continuous for each i (see [10, p. 202]), and An FK-space need not be locally convex, and therein lies the difficulty of the problem since it is easy to see that l1 is the smallest locally convex FK-space.

A metric spaceX = U i-0 x X is constructed such thatX o={x o} consists of a single pointx o , X i , i=0, 1, 2, … are disjoint and closed,X i , i=1, 2, … are open, indX i =0 fori=0, 1, … and indX=1. The above space (proved to be, in some sense, most simple) shows also that the dimension ind of a metric space can be raised by adjoining of a single point, a fact proved recently by E.K. Van Douwen and by T. Przymusinski. Some maximality property of the family {X; IndX=0} is proved and conditions implyingP-ind=P-Ind are given.

Main results: A pseudostratifiable symmetrizable space possesses a symmetric with weak Cauchy condition. A space is strongly symmetrizable if and only if it is a pseudo-open II-image of a metric space. There exists a zero-dimensional symmetrizable space without a symmetric with weak Cauchy condition.

In this paper a relationship is established between the concepts of E-continuity and shape for compact metric spaces, thus answering a question raised by Victor Klee. Victor Klee [4] has asked whether there is a relationship between E-continuity and shape. Here we show that, in the category of compact metric spaces, the shape morphisms can be identified in a one-to-one mariner with the limits of certain equivalence classes of c-continuous functions. Throughout this paper all spaces are assumed to be compact metric spaces. If f is a continuous function (map) from one space X to another space Y, [/] will denote the homotopy equivalence class of f. By Cov(Y) we will denote the set of all finite open covers of Y directed by refinement. If A and B are collections of subsets of Y, A 0, if and only if it is a-continuous for some a E Cov(Y) consisting of open balls of radius E). If g is another function from X to Y, f and g are a-homotopic if there is an a-continuous function H from X x I to Y such that, for each x E X, H(x, 0) = f(x) and H(x, 1) = g(x). This defines an equivalence relation on the set of a-continuous functions from X to Y. We say that f and g are a-close if /f(x), g(x) }: x E X I < a. For a E Cov(Y), [X, Y]a denotes the set of a-homotopy classes of a-continuous functions f from X to Y. The equivalence class of such an f is denoted f] a. By lim [X, Y] a we denote the usual representation, in the category of sets and functions, of the limit of the inverse system if X, Y]a P aal a < a E Cov(Y)} where Paa'([f]a') = [f]a for a < a E Cov(Y). Received by the editors June 29, 1973 and, in revised form, November 11, 1973. AMS (MOS) subject classifications (1970). Primary 55D99.

If X is a compact Hausdorff space we denote by S(X), and by C(X), the hyperspaces of X consisting of all non-empty closed sets, and all non-empty connected closed sets. The topology in each case is the finite topology of Michael ((6), Definition 1·7), in which a sub-base for the open sets is taken consisting of all sets of either of the forms {F|F ⊂ G} and {F|F ∩ G ≠ φ} (where G is any open set of X). Michael has shown that S(X) is also compact Hausdorff ((6), Theorem 4·9·6), and S(X) contains in an obvious way sets which are homeomorphic with C(X) and X itself. We recall that if Xis also a metric space, the topology induced on S(X) (and on C(X)) by Hausdorff's metric is the same as the finite topology ((6), Proposition 3·6).

We analyze the various classes of local irreducible representations (both finite and infinite‐dimensional) of O(3) realized on a linear complex vector space. The well‐known finite‐dimensional unitary representations are obtained when the scalar product is positive definite. The infinite‐dimensional representations are realized on an indefinite metric space. The possible applications of these new classes of representations are briefly discussed.

For Q-spaces (also called functionally closed or Hunt spaces) there are defined in this paper two new invariants, the q-weight and the q*-weight. With the aid of these the following results are obtained. Theorem 1. If τ is a nonmeasurable cardinal number and X is a metric space of weight not exceeding τ, then X is homeomorphic to a closed subspace of the product of copies of a real line R (i.e. ). Theorem 2. If τ is a nonmeasurable cardinal number and X is a complete uniform space whose uniform and topological weights do not exceed τ, then X is homeomorphic to a closed subspace of the product of copies of the real line. Theorem 3. Let X be paracompact, bX a Hausdorff compactification of X, and τ a nonmeasurable cardinal number such that the weight of X does not exceed τ and X is the intersection of a family of not more than τ open subsets of bX. Then X is homeomorphic to a closed subspace of the product of copies of the real line. Bibliography: 8 items.

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.

Let xt be a Markov process on a locally compact metric space (8, p) with a countable base. Let 63 be the o-algebra generated by the open sets, and for ACG6, t>O, set P(t, x, A)=Pr{xtCAjxo=x}. Khasminskii [1 ] has related, in a natural way, the existence of a finite invariant measure for xt with the mean time to hit a compact set. Our object is to relate the existence of such a measure to properties of the measures P(t, x, * ), t > O, xEl. We assume that for fixed t and A, P(t, x, A) is continuous in x, that for an e-neighborhood Sf(x) about x, P(t, x, S,(x))-l as t--O uniformly on compacta, and that P(t, x, 8) = 1. Let C+ be the strictly positive cone of the Banach space ca(&, 63) consisting of the countably additive set functions on 63 with variation norm. Let Ut, t _ 0, be the semigroup defined for M&ca(&, G3) by the formula

In this paper the solutions of the Einstein‐Maxwell equations are investigated under the assumption that the metric of the space‐time and the electromagnetic field are invariant under the transformations of a four‐parametric, simply transitive group.The results can be summarized as follows: In the case of null electromagnetic fields there are two different possibilities; If Λ = 0, all the solutions are Robinson waves; if Λ ≠ 0, there exists only one solution, first given here by (6.26). There exist no other solutions for null electromagnetic fields. In the case of nonnull electromagnetic fields two solutions are found. One metric is known having been first given by Robinson; we give a new solution of type I. The question as to whether there are solutions different from these remains open.

If ε > 0, a subset M of a metric space is said to be ε-connected if for each pair p, q ∈ M there is a finite sequence a0, …, an such that each ai ∈ M, a0 = ρ an = q and the distance from ai−1 to ai is less than ε whenever 0 < i ≦n. It is known [1, p. 117, Satz 1] that a compact metric space is connected if and only if for each ε > 0 it is ε-connected. We present here a proof of an analogous characterization of locally connected unicoherent compacta.

In a previous paper [7], we studied the structure of HID spaces. In this paper, we consider the behavior of HID spaces under monotone mappings. The principal result of this paper is that, given an arbitrary compact metric space Y, there is an HID space X and a monotone map f: X-) Y. We also show that an arbitrary HID space can be mapped monotonically onto a space of any preassigned dimension, and that, given an HID space X and a positive integer n, there is an n-dimensional space Y and a monotone map f: Y->X. R. H. Bing showed in [2 ] that each nondegenerate monotone image of a pseudo-arc is a pseudo-arc. The results of this paper show that no similar monotone invariance property holds for spaces of dimension greater than 1. In this paper, all spaces will be compact metric spaces (compacta). We will be dealing with the Hilbert cube, which we regard as being the product of a countably infinite collection of straight line intervals IX = 11 X 12 X 13 X ... , where Ij = [-1/2i, 1/2'].

Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.

If (X, d) is a metric space, a homeomorphism$ : X + X is called expansive if there exists a S > 0 (called an expansive constant) such that J(+, C#PY) 0 and a closed subset E of the product space nyEey, (0, l,..., h 1) which is invariant under the shift homeomorphism u and such that the given expansive homeomorphism is a quotient of o i E). When X is O-dimensional 4 is conjugate to a subshift. Also the topological entropy h(4) f o an expansive homeomorphism + is related to the periodic points of (5 by the formula

A necessary and sufficient condition of correctness of extremal problems for lower semicontinuous functionals defined in metric spaces is given.