Almost Continuity of Mappings

1968 ◽  
Vol 11 (3) ◽  
pp. 453-455 ◽  
Author(s):  
Shwu-Yeng T. Lin
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Direct Proof ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Natural Context ◽  
Set Of Points

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].

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

ON THE STEINER RATIO IN $\mathcal{R}_{n}$

2011 ◽  
Vol 03 (04) ◽  
pp. 473-489
Author(s):  
HAI DU ◽  
WEILI WU ◽  
ZAIXIN LU ◽  
YINFENG XU
Keyword(s):  
Euclidean Space ◽  
Spanning Tree ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Minimum Length ◽  
Steiner Ratio ◽  
Line Segments ◽  
Finite Set ◽  
Steiner Minimum Tree ◽  
Set Of Points

The Steiner minimum tree and the minimum spanning tree are two important problems in combinatorial optimization. Let P denote a finite set of points, called terminals, in the Euclidean space. A Steiner minimum tree of P, denoted by SMT(P), is a network with minimum length to interconnect all terminals, and a minimum spanning tree of P, denoted by MST(P), is also a minimum network interconnecting all the points in P, however, subject to the constraint that all the line segments in it have to terminate at terminals. Therefore, SMT(P) may contain points not in P, but MST(P) cannot contain such kind of points. Let [Formula: see text] denote the n-dimensional Euclidean space. The Steiner ratio in [Formula: see text] is defined to be [Formula: see text], where Ls(P) and Lm(P), respectively, denote lengths of a Steiner minimum tree and a minimum spanning tree of P. The best previously known lower bound for [Formula: see text] in the literature is 0.615. In this paper, we show that [Formula: see text] for any n ≥ 2.

scholarly journals The Borel structure of iterates of continuous functions

1989 ◽  
Vol 32 (3) ◽  
pp. 483-494 ◽  
Author(s):  
Paul D. Humke ◽  
M. Laczkovich
Keyword(s):  
Banach Space ◽  
Continuous Functions ◽  
Baire Space ◽  
Space Of Continuous Functions ◽  
Borel Structure ◽  
Image Position ◽  
Set Of Points ◽  
Positive Integers

Let C[0,1] be the Banach space of continuous functions defined on [0,1] and let C be the set of functions f∈C[0,1] mapping [0,1] into itself. If f∈C, fk will denote the kth iterate of f and we put Ck = {fk:f∈C;}. The set of increasing (≡ nondecreasing) and decreasing (≡ nonincreasing) functions in C will be denoted by ℐ and D, respectively. If a function f is defined on an interval I, we let C(f) denote the set of points at which f is locally constant, i.e.We let N denote the set of positive integers and NN denote the Baire space of sequences of positive integers.

The anomaly of convex bodies

1953 ◽  
Vol 49 (1) ◽  
pp. 54-58 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Real Numbers

I write X for the point (x1, x2, …, xn) of n-dimensional Euclidean space Rn. The coordinates x1, x2, …, xn are real numbers. The origin (0, 0,…, 0) is denoted by O. If t is a real number, tX denotes the point (tx1, tx2, …, txn); in particular, − X is the point (−x1, −x2,…, −xn). Also X + Y denotes the point {x1 + y1, x2 + y2, …, xn + yn).

scholarly journals Topological Manifolds

2014 ◽  
Vol 22 (2) ◽  
pp. 179-186 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Euclidean Space ◽  
Topological Space ◽  
Interior Point ◽  
Cartesian Product ◽  
Closed Ball ◽  
Topological Manifold ◽  
Open Ball ◽  
Connected Component ◽  
Euclidean Spaces ◽  
Topological Manifolds

Summary Let us recall that a topological space M is a topological manifold if M is second-countable Hausdorff and locally Euclidean, i.e. each point has a neighborhood that is homeomorphic to an open ball of E n for some n. However, if we would like to consider a topological manifold with a boundary, we have to extend this definition. Therefore, we introduce here the concept of a locally Euclidean space that covers both cases (with and without a boundary), i.e. where each point has a neighborhood that is homeomorphic to a closed ball of En for some n. Our purpose is to prove, using the Mizar formalism, a number of properties of such locally Euclidean spaces and use them to demonstrate basic properties of a manifold. Let T be a locally Euclidean space. We prove that every interior point of T has a neighborhood homeomorphic to an open ball and that every boundary point of T has a neighborhood homeomorphic to a closed ball, where additionally this point is transformed into a point of the boundary of this ball. When T is n-dimensional, i.e. each point of T has a neighborhood that is homeomorphic to a closed ball of En, we show that the interior of T is a locally Euclidean space without boundary of dimension n and the boundary of T is a locally Euclidean space without boundary of dimension n − 1. Additionally, we show that every connected component of a compact locally Euclidean space is a locally Euclidean space of some dimension. We prove also that the Cartesian product of locally Euclidean spaces also forms a locally Euclidean space. We determine the interior and boundary of this product and show that its dimension is the sum of the dimensions of its factors. At the end, we present several consequences of these results for topological manifolds. This article is based on [14].

scholarly journals A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION

2021 ◽  
Vol 9 (1) ◽  
pp. 250-263
Author(s):  
V. Mykhaylyuk ◽  
O. Karlova
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Baire Space ◽  
Regular Space ◽  
Topological Spaces ◽  
Infinite Cardinal ◽  
Urysohn Space ◽  
Completely Regular

In 1932 Sierpi\'nski proved that every real-valued separately continuous function defined on the plane $\mathbb R^2$ is determined uniquely on any everywhere dense subset of $\mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $\mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:X\times Y\to \mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $X\times Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $X\times Y$. Henriksen and Woods proved that for an infinite cardinal $\aleph$, an $\aleph^+$-Baire space $X$ and a topological space $Y$ with countable $\pi$-character every separately continuous function $f:X\times Y\to \mathbb R$ is also determined uniquely on everywhere dense subset of $X\times Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $\pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $\pi$-character, $Z$ is Urysohn space, $A\subseteq X\times Y$ is everywhere dense set, $f:X\times Y\to Z$ and $g:X\times Y\to Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi\'nsi-type theorems.

scholarly journals On the complexity of some partition problems of a finite set of points in Euclidean space into balanced clusters

2019 ◽  
Vol 488 (1) ◽  
pp. 16-20
Author(s):  
A. V. Kel’manov ◽  
A. V. Pyatkin ◽  
V. I. Khandeev
Keyword(s):  
Euclidean Space ◽  
Cluster Size ◽  
Np Completeness ◽  
Finite Set ◽  
Set Of Points

We consider some problems of partitioning a finite set of N points in d-dimension Euclidean space into two clusters balancing the value of (1) the quadratic variance normalized by a cluster size, (2) the quadratic variance, and (3) the size-weighted quadratic variance. We have proved the NP-completeness of all these problems.

Minkowski spacetime

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Euclidean Space ◽  
Special Relativity ◽  
Three Dimensional ◽  
Minkowski Spacetime ◽  
Dimensional Euclidean Space ◽  
Fourth Dimension ◽  
Relativistic Dynamics ◽  
Newtonian Theory ◽  
Space And Time ◽  
Set Of Points

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical extension of three-dimensional Euclidean space. In special relativity, ‘relative, apparent, and common’ (in the words of Newton) space and time are represented by a mathematical set of points called events, which constitute the Minkowski spacetime. This chapter also stresses the interpretation of the fourth dimension, which in special relativity is time. Here, time now loses the ‘universal’ and ‘absolute’ nature that it had in the Newtonian theory.

Self-Centred Sets

1966 ◽  
Vol 18 ◽  
pp. 974-980
Author(s):  
H. Kestelman
Keyword(s):  
Abelian Group ◽  
Trigonometric Series ◽  
Absolute Convergence ◽  
Rational Numbers ◽  
Real Numbers ◽  
Space Group ◽  
Space Groups ◽  
Set Of Points

A subset S of an abelian group G is said to have a centre at a if whenever x belongs to S so does 2a — x. This note is mainly concerned with self-centred sets, i.e. those S with the property that every element of S is a centre of S. Such sets occur in the study of space groups: the set of inversion centres of a space group is always self-centred. Every subgroup of G is self-centred, so is every coset in G: this is the reason why the set of points of absolute convergence of a trigonometric series is self-centred or empty (1). A self-centred set of real numbers that is either discrete or consists of rational numbers must in fact be a coset (see §3); this does not hold for an arbitrary enumerable self-centred set of real numbers (§3.3).

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