Semi-metric Space Research Articles

Common fixed point theorems for compatible self-maps of Hausdorff topological spaces

AbstractThe concept of proper orbits of a map "Equation missing" is introduced and results of the following type are obtained. If a continuous self-map "Equation missing" of a Hausdorff topological space "Equation missing" has relatively compact proper orbits, then "Equation missing" has a fixed point. In fact, "Equation missing" has a common fixed point with every continuous self-map "Equation missing" of "Equation missing" which is nontrivially compatible with "Equation missing". A collection of metric and semimetric space fixed point theorems follows as a consequence. Specifically, a theorem by Kirk regarding diminishing orbital diameters is generalized, and a fixed point theorem for maps with no recurrent points is proved.

Nonparametric models for functional data, with application in regression, time series prediction and curve discrimination

The aim of this article is to investigate a new approach for estimating a regression model with scalar response and in which the explanatory variable is valued in some abstract semi-metric functional space. Nonparametric estimates are introduced, and their behaviors are investigated in the situation of dependent data. Our study contains asymptotic results with rates. The curse of dimensionality, which is of great importance in this infinite dimensional setting, is highlighted by our asymptotic results. Some ideas, based on fractal dimension modelizations, are given to reduce dimensionality of the problem. Generalization of the model leads to possible applications in several fields of applied statistics, and we present three applications among these namely: regression estimation, time-series prediction, and curve discrimination. As a by-product of our approach in the finite-dimensional context, we give a new proof for the rates of convergence of some Nadaraya–Watson kernel-type smoother without needing any smoothness assumption on the density function of the explanatory variables.

Estimation non paramétrique de la régression avec variable explicative dans un espace métrique

We study a nonparametric regression estimator when the explanatory variable takes its values in a semi-metric space. We establish some asymptotic results and give upper bounds of the p-mean and the almost sure estimation errors under general conditions. We end by an application to the discrimination in a semi-metric space and illustrate the results by the example of Wiener process as an explanatory variable. To cite this article: S. Dabo-Niang, N. Rhomari, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Natural entropies in the state space of damaged bodies

The possibility of the introduction of natural entropies without probability in the semimetric state space of damaged bodies is investigated. Lebesgue lemma about covers is extended to this space and its consequences on entropies of homomorphisms (i.e. entropies of state transformations) are analyzed.

Cosmicity of cometrizable spaces

A space X X is cometrizable if X X has a coarser metric topology such that each point of X X has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space X X : (a) X X is the continuous image of a separable metric space; (b) X ω {X^\omega } is hereditarily separable and hereditarily Lindelöf, (c) X 2 {X^2} has no uncountable discrete subspaces; (d) X X is a Lindelöf semimetric space; (e) X X has the pointed ccc {\text {ccc}} . This result is a corollary to our main result which states that, under PFA, if X X is a cometrizable space with no uncountable discrete subspaces, then either X X is the continuous image of a separable metric space or X X contains a copy of an uncountable subspace of the Sorgenfrey line.

Metric transforms of finite spaces and connected graphs

A metric transform of a semimetric space X is obtained from X by measuring the distances by a different (not always proportional) scale. Two semimetric spaces are said to be isomorphic if one is isometric to a metric transform of the other. If X is a finite semimetric space, then it will be shown that X is isomorphic to a subset of a euclidean space. The dimension of X is defined to be the minimum dimension of a euclidean space containing an isomorph of X. In this paper we examine scales and dimensions for finite semimetric spaces, especially, for connected graphs and trees as metric spaces. We also count the number of non-isomorphic semimetric spaces.

The probability of the triangle inequality in probabilistic metric squares

The notion of a random semi-metric space provides an alternate approach to the study of probabilistic metric spaces from the standpoint of random variables instead of distribution functions and permits a new investigation of the triangle inequality. Starting with a probability space (Ω,B,P) and an abstract setS, each pair of points,p, q, inS is assigned a random variableX pq with the interpretation thatX pq (ω) is the distance betweenp andq at the “instant” ω. The probability of the eventJ pqr = {ω ∈ Ω:X pr (ω)≤X pq (ω)+X qr (ω)} is studied under distribution function conditions imposed by Menger Spaces (K. Menger, “Statistical Metrics,” Proc. Nat. Acad. Sci., U.S.A., 28 (1942), 535–537; B. Schweizer and A. Sklar, “Statistical Metric Spaces,” Pacific J. Math.10 (1960), 313–334). It turns out that for e > 0 there are 3 non-negative, identically-distributed random variablesX, Y andZ for whichP(X <Y +Z) < e. This and other results show that distribution function triangle inequalities are very weak. Conditional probabilities are introduced to give necessary and sufficient conditions forP(J pqr ) = 1.

On the linear connection and curvature in Newtonian mechanics

The trajectories of a scleronomic, holonomic particle motion in an otherwise general force field are autoparallel curves in a linear connected, symmetric, ’’almost’’ semimetric space. The Riemann–Christoffel curvature tensor and its concomitants belonging to the dynamical affinity are defined, and the physical meaning is discussed.

Spaces with subparacompact completions

If the completely regular space X admits a quasi-perfect map onto the Dieudonné complete semi-metric space Y, then δX, the Dieudonné completion of X, is subparacompact; if in addition, Y is a Moore space, then δX is a subparacompact p-space.

A Regular Lindelof Semimetric Space which has no Countable Network

A Regular Lindelof Semimetric Space which has no Countable Network

A regular Lindelöf semimetric space which has no countable network

A completely regular semimetric space M M is constructed which has no σ \sigma -discrete network. The space M M constructed has the property that every subset of M M of cardinality 2 ℵ 0 {2^{{\aleph _0}}} contains a limit point of itself; thus, assuming 2 ℵ 0 = ℵ 1 , M {2^{{\aleph _0}}} = {\aleph _1},M is Lindelöf. It is also shown from the same space M M that, assuming 2 ℵ 0 = ℵ 1 {2^{{\aleph _0}}} = {\aleph _1} , there exists a regular Lindelöf semimetric space X X such that X × X X \times X is not normal (hence not Lindelöf).

A Paracompact Semi-Metric Space Which is Not an M 3 -Space

A Paracompact Semi-Metric Space Which is Not an M 3 -Space

A paracompact semi-metric space which is not an $M\sb{3}$-space

E. A. Michael has shown that a regular Hausdorff space 5 is paracompact if and only if any of the following is true: (1) every open cover of 5 has a a-locally finite open refinement [8], (2) every open cover of 5 has an open ^-closure preserving refinement [9], or (3) every open cover of S has an open <r-cushioned refinement [lO]. The Nagata-Smirnov Theorem [12] or [14] (see also Bing's Theorem 3 in [l]) states that a P3-space with a <r-locally finite base is metrizable. In [3] Jack Ceder defines an Mi-space to be a P3-space with a cr-closure preserving base and an M3-space to be a Pi-space with a <r-cushioned pair base (see Definition 1 below). By Michael's theorems cited above both Miand Af3-spaces are paracompact, and in view of the Nagata-Smirnov Theorem, one might suspect that they would even be metrizable. In [3], however, Ceder shows that such is not the case; in fact Miand M3-spaces need not be first countable, and they need not be metrizable even if they are first countable. Ceder does show, though, that a first countable M3(and hence Mi-) space is a paracompact semi-metric space, and he raises the question: is this a characterization—i.e. is every paracompact semi-metric space an Af3-space (and perhaps even an Mi-space)? A negative answer is given to that question in this paper. Also it was called to my attention by Professor Michael that a question raised by C. Borges [2] is answered (in the negative) by the same example below, which is a cosmic space (the regular continuous image of a separable metric space [ll]) that is not an Af3-space. Definition 1. Let P be a collection of ordered pairs of subsets of the TV-space S such that, for each p = (pi, p2)EP, px is open and P1C.P2, and such that, for every xES and every neighborhood U of x, there is a pEP for which XEP1C.P2EU. Then P is called a pair base for 5. Moreover, P is called cushioned if, for every QQP,

Axioms that define semi-metric, Moore and metric spaces

In [1 I L. F. McAuley asked the following question: is it possible to partition . . Moore's metrization theorem into three or more parts which begins with a condition for a topological space and which ends with a condition for a metrizable space, but with necessary and sufficient conditions somewhere between these extremes for semi-metric and Moore spaces? Z, stated below, is such a partitioning. The notation Axiom Zi denotes parts (1), (2), * * *, (i) of Z. In ?1 it is proved in Theorems 1, 2, and 3, respectively, that a necessary and sufficient condition for a topological space to be semi-metrizable, a Moore space, and metrizable is that it satisfy Z2, Z3, and Z4 respectively. A counter-example is given in ?2 which shows that the argument for the statement a Moore space is a semimetric topological space in Theorem 6.2 in [I I is not correct. Finally, in ?3 it is shown that part (3) of Theorem 2 in [2 ] can be changed so that the resulting statement is equivalent to a Moore space. Definitions are given in [I ]. DEFINITION. If { Jn } denotes a sequence such that for each natural number n, Jn denotes a collection of neighborhoods covering a point set M, then the sequence I Bi }, where i denotes a natural number, is said to be a basic refinement of { Jn } for M provided that with each point p in M there is associated a sequence I bi(p) } such that for each i: (1) bi(p) is a neighborhood in J, }, (2) bi+1(p) is a subset of b,(p), (3) p is the only point common to I bi(p) }, and (4) Bi denotes the collection of all neighborhoods bi(p) for all points in M.

A Regular Semi-Metric Space for which There is no Semi-Metric Under which all Spheres are Open

A Regular Semi-Metric Space for which There is no Semi-Metric Under which all Spheres are Open

A regular semi-metric space for which there is no semi-metric under which all spheres are open

A regular semi-metric space for which there is no semi-metric under which all spheres are open

A note on complete collectionwise normality and paracompactness

1. A question that has aroused considerable interest and which has remained unanswered is the following. Is a normal Moore space metrizable? Both R. H. Bing and F. B. Jones have important results which go a long way toward answering this question. For example, Bing has proved that a collectionwise normal Moore space is metrizable [l ] while Jones has shown that a separable normal Moore space is metrizable [2] provided the continuum hypothesis holds true. Although the results given in the present paper fail to answer the question at hand, it is shown that certain types of normality and paracompactness are equivalent in a Moore space, indeed, in more abstract spaces such as a semimetric topological space.2 2. Complete collectionwise normality. The reader is referred to R. H. Bing's paper [l] for definitions concerning screenability, collectionwise normality, discrete collections, and related concepts. We say that a collection G is almost discrete if and only if it is discrete with respect to G* (the logical sum of the elements of G). Furthermore, a space is completely collectionwise normal if and only if for each almost discrete collection G of point sets, there is a collection H of disjoint open sets covering G* such that no element of H intersects two elements of G. A space is hereditarily collectionwise normal if and only if each subspace is collectionwise normal. It is not difficult to show that complete collectionwise normality is equivalent to hereditary collectionwise normality in a topological space. A topological space 5 is said to be F,-screenable if and only if for each open covering G of S, there exists a sequence {Xi} such that (1) for each i, Xi is a discrete collection of closed point sets each of which lies in an element of G and (2) E-^«' covers S. There is an example [5] of a Hausdorff space satisfying the first axiom of count

A remark on M. M. Day’s characterization of inner-product spaces and a conjecture of L. M. Blumenthal

1. A space of elements a, b, ■ • • , with a distance function ab is said to be semi-metric provided ab = ba>0 if a^b, and aa = 0. A reallinear space of elements/, g, ■ • ■ is said to be semi-normed provided a function 11/11 is defined in S having the usual properties of a norm with the exception of the inequality ||/+g|| =||/||+||g||> which is not assumed. Evidently ||/—g is a semimetric in the sense of the first definition. A semimetric space is called ptolemaic provided that among the distances between any four points a, b, c, d Ptolemy's inequality