scholarly journals The Metrizability of Spaces whose Diagonals have a Countable Base

1977 ◽  
Vol 20 (4) ◽  
pp. 513-514 ◽  
Author(s):  
John Ginsburg
Keyword(s):  
Metrizable Space ◽  
Countable Base ◽  
Neighborhood Base ◽  
Isolated Points

AbstractIt is shown that the diagonal of X has a countable neighborhood base in X × X if and only if X is a metrizable space whose set of non-isolated points is compact.

scholarly journals Regular bases at non-isolated points and metrization theorems

2012 ◽  
Vol 49 (1) ◽  
pp. 91-105 ◽  
Author(s):  
Fucai Lin ◽  
Shou Lin ◽  
Heikki Junnila
Keyword(s):  
Affirmative Answer ◽  
Metrizable Space ◽  
Regular Space ◽  
Countable Base ◽  
Locally Finite ◽  
Finite Base ◽  
Isolated Points ◽  
Metrizable Spaces ◽  
Perfect Space

In this paper, we define the spaces with a regular base at non-isolated points and discuss some metrization theorems. We firstly show that a space X is a metrizable space, if and only if X is a regular space with a σ-locally finite base at non-isolated points, if and only if X is a perfect space with a regular base at non-isolated points, if and only if X is a β-space with a regular base at non-isolated points. In addition, we also discuss the relations between the spaces with a regular base at non-isolated points and some generalized metrizable spaces. Finally, we give an affirmative answer for a question posed by F. C. Lin and S. Lin in [7], which also shows that a space with a regular base at non-isolated points has a point-countable base.

scholarly journals A function space from a compact metrizable space to a dendrite with the hypo-graph topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Hanbiao Yang ◽  
Katsuro Sakai ◽  
Katsuhisa Koshino
Keyword(s):  
Finite Number ◽  
Function Space ◽  
Metrizable Space ◽  
Vietoris Topology ◽  
Hilbert Cube ◽  
Graph Topology ◽  
Closed Sets ◽  
Continuous Map ◽  
Compact Metrizable Space ◽  
Isolated Points

Abstract Let X be an infinite compact metrizable space having only a finite number of isolated points and Y be a non-degenerate dendrite with a distinguished end point v. For each continuous map ƒ : X → Y , we define the hypo-graph ↓vƒ = ∪ x∈X {x} × [v, ƒ (x)], where [v, ƒ (x)] is the unique arc from v to ƒ (x) in Y . Then we can regard ↓v C(X, Y ) = {↓vƒ | ƒ : X → Y is continuous} as the subspace of the hyperspace Cld(X × Y ) of nonempty closed sets in X × Y endowed with the Vietoris topology. Let be the closure of ↓v C(X, Y ) in Cld(X ×Y ). In this paper, we shall prove that the pair , ↓v C(X, Y )) is homeomorphic to (Q, c0), where Q = Iℕ is the Hilbert cube and c0 = {(xi )i∈ℕ ∈ Q | limi→∞xi = 0}.

scholarly journals Some remarks on quasi-uniform spaces

1989 ◽  
Vol 31 (3) ◽  
pp. 309-320 ◽  
Author(s):  
Hans-Peter A. Künzi
Keyword(s):  
Topological Space ◽  
Compact Space ◽  
Metrizable Space ◽  
Main Step ◽  
Countable Base ◽  
Uniform Spaces

A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

A Hilbert cube compactification of the function space with the compact-open topology

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Atsushi Kogasaka ◽  
Katsuro Sakai
Keyword(s):  
Closed Interval ◽  
Continuous Functions ◽  
Metrizable Space ◽  
Hilbert Cube ◽  
Compact Open Topology ◽  
Locally Compact ◽  
Closed Sets ◽  
Isolated Points ◽  
Compact Case ◽  
Separable Metrizable Space

AbstractLet X be an infinite, locally connected, locally compact separable metrizable space. The space C(X) of real-valued continuous functions defined on X with the compact-open topology is a separable Fréchet space, so it is homeomorphic to the psuedo-interior s = (−1, 1)ℕ of the Hilbert cube Q = [−1, 1]ℕ. In this paper, generalizing the Sakai-Uehara’s result to the non-compact case, we construct a natural compactification $$ \bar C $$(X) of C(X) such that the pair ($$ \bar C $$(X), C(X)) is homeomorphic to (Q, s). In case X has no isolated points, this compactification $$ \bar C $$(X) coincides with the space USCCF(X,) of all upper semi-continuous set-valued functions φ: X → = [−∞, ∞] such that each φ(x) is a closed interval, where the topology for USCCF(X, ) is inherited from the Fell hyperspace Cld*F(X × ) of all closed sets in X × .

Markov spectrum near Freiman's isolated points in M ∖ L

2019 ◽  
Vol 194 ◽  
pp. 390-408 ◽  
Author(s):  
Carlos Matheus ◽  
Carlos Gustavo Moreira
Keyword(s):  
Isolated Points ◽  
Markov Spectrum

Functional Data Analyses for the Assessment of Joint Power Profiles During Gait of Stroke Subjects

2014 ◽  
Vol 30 (2) ◽  
pp. 348-352 ◽  
Author(s):  
André G. P. Andrade ◽  
Janaine C. Polese ◽  
Leopoldo A. Paolucci ◽  
Hans-Joachim K. Menzel ◽  
Luci F. Teixeira-Salmela
Keyword(s):  
Power Generation ◽  
Gait Cycle ◽  
Sagittal Plane ◽  
Plantar Flexion ◽  
Knee Extension ◽  
Mathematical Function ◽  
Joint Power ◽  
Gait Biomechanics ◽  
Isolated Points ◽  
Parametric Analyses

Lower extremity kinetic data during walking of 12 people with chronic poststroke were reanalyzed, using functional analysis of variance (FANOVA). To perform the FANOVA, the whole curve is represented by a mathematical function, which spans the whole gait cycle and avoids the need to identify isolated points, as required for traditional parametric analyses of variance (ANOVA). The power variables at the ankle, knee, and hip joints, in the sagittal plane, were compared between two conditions: With and without walking sticks at comfortable and fast speeds. For the ankle joint, FANOVA demonstrated increases in plantar flexion power generation during 60–80% of the gait cycle between fast and comfortable speeds with the use of walking sticks. For the knee joint, the use of walking sticks resulted in increases in the knee extension power generation during 10–30% of the gait cycle. During both speeds, the use of walking sticks resulted in increased power generation by the hip extensors and flexors during 10–30% and 40–70% of the gait cycle, respectively. These findings demonstrated the benefits of applying the FANOVA approach to improve the knowledge regarding the effects of walking sticks on gait biomechanics and encourage its use within other clinical contexts.

scholarly journals Infinite Powers and Cohen Reals

2018 ◽  
Vol 61 (4) ◽  
pp. 812-821 ◽  
Author(s):  
Andrea Medini ◽  
Jan van Mill ◽  
Lyubomyr Zdomskyy
Keyword(s):  
Open Problem ◽  
Metrizable Space ◽  
Almost Everywhere ◽  
Separable Metrizable Space ◽  
Insight Into

AbstractWe give a consistent example of a zero-dimensional separable metrizable space Z such that every homeomorphism of Zω acts like a permutation of the coordinates almost everywhere. Furthermore, this permutation varies continuously. This shows that a result of Dow and Pearl is sharp, and gives some insight into an open problem of Terada. Our example Z is simply the set of ω1 Cohen reals, viewed as a subspace of 2ω.

scholarly journals Monstars on Glaciers

1983 ◽  
Vol 29 (101) ◽  
pp. 70-77 ◽  
Author(s):  
J. F. Nye
Keyword(s):  
Numerical Simulation ◽  
Strain Rate ◽  
Tensor Field ◽  
Pure Shear ◽  
Glacier Surface ◽  
Isotropic Point ◽  
Different Types ◽  
Isolated Points ◽  
Principal Directions ◽  
Structurally Stable

AbstractIsotropic points are structurally stable features of any complicated field of stress or strain-rate, and therefore will almost always be present on the surface of a glacier. A given isotropic point for strain-rate will belong to one of six different classes, depending on the pattern (lemon, star, or monstar) of principal directions and the contours (ellipses or hyperbolas) of constant principal strain-rate values in its neighbourhood. The central isotropic point on a glacier should theoretically have a monstar pattern, but the contours around it may sometimes be elliptic and sometimes hyperbolic. Nearby, but not coincident with it there will be an isotropic point for stress. This will also have a monstar pattern but, in contrast to the strain-rate point, the contours around it must be hyperbolic. Published examples are consistent with these conclusions. In addition to isotropic points for strain-rate a glacier surface will contain isolated points of pure shear; these also can be classified into six different types. Stable features of this kind give information about the essential structure of a tensor field and form useful points of comparison between observation and numerical simulation.

scholarly journals Ends of locally compact groups and their coset spaces

1974 ◽  
Vol 17 (3) ◽  
pp. 274-284 ◽  
Author(s):  
C. H. Houghton
Keyword(s):  
Compact Group ◽  
Direct Product ◽  
Compact Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Countable Base ◽  
Locally Compact ◽  
Coset Spaces ◽  
Compact Boundary

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

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