Countable Compagtifications

1966 ◽  
Vol 18 ◽  
pp. 616-620 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Positive Integer ◽  
Compact Space ◽  
Real Line ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Natural Numbers ◽  
Finite Sets ◽  
The Real ◽  
One To One

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

scholarly journals THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

2020 ◽  
Vol 6 (2) ◽  
pp. 108
Author(s):  
Tursun K. Yuldashev ◽  
Farhod G. Mukhamadiev
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Real Line ◽  
Local Density ◽  
Cardinal Number ◽  
Topological Spaces ◽  
Topological Properties ◽  
Locally Compact ◽  
The Real ◽  
Weak Density

In this paper, the local density \((l d)\) and the local weak density \((l w d)\) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree \(S P^{n}\) and the subfunctor of permutation degree \(S P_{G}^{n}\),  \(P\) is the cardinal number of topological spaces. Let \(X\) be an infinite \(T_{1}\)-space. We prove that the following propositions hold.(1) Let \(Y^{n} \subset X^{n}\); (A) if \(d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)\), then \(d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)\); (B) if \(l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)\), then \(l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)\). (2) Let \(Y\subset X\); (A) if \(l d \,(Y)=l d \,(X)\), then \(l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)\); (B) if \(w d \,(Y)=w d \,(X)\), then \(w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)\).(3) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is a locally compact \(T_{1}\)-space, then \(S P^{n} X, \, S P_{G}^{n} X\), and \(\exp _{n} X\) are \(k\)-spaces.(4) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is an infinite \(T_{1}\)-space, then \(n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)\).We also have studied that the functors \(SP^{n},\) \(SP_{G}^{n} ,\) and \(\exp _{n} \) preserve any \(k\)-space. The functors \(SP^{2}\) and \(SP_{G}^{3}\) do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite \(T_{1}\)-space \(X\) coincides with the densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\). It is also shown that the weak density of an infinite \(T_{1}\)-space \(X\) coincides with the weak densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\).

scholarly journals Examples of expanding maps with some special properties

1987 ◽  
Vol 36 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Bau-Sen Du
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Periodic Point ◽  
Topological Entropy ◽  
Real Line ◽  
Periodic Points ◽  
Unit Interval ◽  
Expanding Maps ◽  
The Real ◽  
Piecewise Monotonic

Let I be the unit interval [0, 1] of the real line. For integers k ≥ 1 and n ≥ 2, we construct simple piecewise monotonic expanding maps Fk, n in C0 (I, I) with the following three properties: (1) The positive integer n is an expanding constant for Fk, n for all k; (2) The topological entropy of Fk, n is greater than or equal to log n for all k; (3) Fk, n has periodic points of least period 2k · 3, but no periodic point of least period 2k−1 (2m+1) for any positive integer m. This is in contrast to the fact that there are expanding (but not piecewise monotonic) maps in C0(I, I) with very large expanding constants which have exactly one fixed point, say, at x = 1, but no other periodic point.

scholarly journals Wadge-like reducibilities on arbitrary quasi-Polish spaces

2014 ◽  
Vol 25 (8) ◽  
pp. 1705-1754 ◽  
Author(s):  
LUCA MOTTO ROS ◽  
PHILIPP SCHLICHT ◽  
VICTOR SELIVANOV
Keyword(s):  
Real Line ◽  
Polish Space ◽  
Baire Space ◽  
Topological Spaces ◽  
Polish Spaces ◽  
The Real ◽  
Degree Structure ◽  
Wadge Hierarchy

The structure of the Wadge degrees on zero-dimensional spaces is very simple (almost well ordered), but for many other natural nonzero-dimensional spaces (including the space of reals) this structure is much more complicated. We consider weaker notions of reducibility, including the so-called Δ0α-reductions, and try to find for various natural topological spaces X the least ordinal αX such that for every αX ⩽ β < ω1 the degree-structure induced on X by the Δ0β-reductions is simple (i.e. similar to the Wadge hierarchy on the Baire space). We show that αX ⩽ ω for every quasi-Polish space X, that αX ⩽ 3 for quasi-Polish spaces of dimension ≠ ∞, and that this last bound is in fact optimal for many (quasi-)Polish spaces, including the real line and its powers.

On Fourier-Stieltjes Transforms

1955 ◽  
Vol 7 ◽  
pp. 453-461 ◽  
Author(s):  
A. P. Calderón ◽  
A. Devinatz
Keyword(s):  
Real Line ◽  
The Real ◽  
One To One ◽  
Image Position ◽  
Decreasing Functions ◽  
Stieltjes Transforms

Let be the class of bounded non-decreasing functions defined on the real line which are normalized by the conditions ϕ(− ∞) = 0 , ϕ(t + 0) = ϕ(t).Let be the class of Fourier-Stieltjes transforms of elements of i.e. the elements of and are connected by the relationwhere ϕ ∊ and Φ ∊ .It is well known, and easy to verify that this mapping from to is one to one (1, p. 67, Satz 18).

scholarly journals The Diagonalization Paradox

2020 ◽  
Author(s):  
Ron Ragusa
Keyword(s):  
Initial Conditions ◽  
Open Interval ◽  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Diagonal Argument ◽  
One To One

In 1891 Georg Cantor published his Diagonal Argument which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1).

scholarly journals The Diagonalization Paradox - Expanded

2020 ◽  
Author(s):  
Ron Ragusa
Keyword(s):  
Initial Conditions ◽  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
One To One ◽  
The One

In 1891 Georg Cantor published his Diagonal Method which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of Cantor’s proof we can use the diagonal method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the interval (0, 1). In the appendix we demonstrate that using the diagonal method recursively will, at the limit of the process, fully account for all the infinite binary decimals in (0, 1). The proof will cement the one-to-one correspondence between the natural numbers and the infinite binary decimals in (0, 1).

scholarly journals On compact one-to-one continuous images of the real line

1971 ◽  
Vol 23 (2) ◽  
pp. 251-256 ◽  
Author(s):  
Anatole Beck ◽  
Jonathan Lewin ◽  
Mirit Lewin
Keyword(s):  
Real Line ◽  
The Real ◽  
One To One ◽  
Continuous Images

scholarly journals Sets and Subseries

1957 ◽  
Vol 9 ◽  
pp. 223-224
Author(s):  
G. M. Petersen
Keyword(s):  
Real Line ◽  
Line Segment ◽  
Binary Expansion ◽  
Original Series ◽  
The Real ◽  
One To One

Suppose αn ≠ 0 for all n. By replacing a set of terms from the series Σαn by 0's, we form a subseries of the original series. These subseries can be put in one to one correspondence with the non-terminating binary expansion of the points on the real line segment (0, 1).

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