scholarly journals Joins of topologies homeomorphic to the rationals

1989 ◽  
Vol 47 (2) ◽  
pp. 256-262
Author(s):  
A. J. Jayanthan ◽  
V. Kannan
Keyword(s):  
Topological Space ◽  
Rational Numbers ◽  
Countably Infinite

AbstractLet Q be the space of all rational numbers and (X, τ) be a topological space where X is countably infinite. Here we prove that (1) τ is the join of two topologies on X both homeomorphic to Q if and only if τ is non-compact and metrizable, and (2) τ is the join of topologies on X each homeomorphic to Q if and only if τ is Tychonoff and noncompact.

A note on lemmas of Green and Kondo

1967 ◽  
Vol 63 (1) ◽  
pp. 83-85 ◽  
Author(s):  
A. O. Morris
Keyword(s):  
Symmetric Functions ◽  
Rational Numbers ◽  
Infinite Sets ◽  
Image Position ◽  
Countably Infinite

Let R be the field of rational numbers, {x} = {x1, z2, …}, {y} = {y1, y2, …} be two countably infinite sets of variables and t an indeterminate. Let (λ) = (λ1, λ2, …, λm) be a partition of n. Then Littlewood (5) has shown thatcan be expressed in the formwhere Qλ(x, t) and Qλ(y, t) denote certain symmetric functions on the sets {x} and {y} respectively. In additionwhere is the partition of n conjugate to (λ). In fact, Littlewood (5) showed thatwhere the summation is over all terms obtained by permutations of the variables xi (i = 1, 2, …) and.

scholarly journals Semigroups of continuous selfmaps for which Green's and ℐ relations coincide

1979 ◽  
Vol 20 (1) ◽  
pp. 25-28 ◽  
Author(s):  
K. D. Magill
Keyword(s):  
Topological Space ◽  
Metric Spaces ◽  
Transformation Semigroup ◽  
Doctoral Dissertation ◽  
Discrete Space ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
The One ◽  
One Point Compactification ◽  
Countably Infinite

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following

Countable structures, Ehrenfeucht strategies, and Wadge reductions

1991 ◽  
Vol 56 (4) ◽  
pp. 1325-1348 ◽  
Author(s):  
Tom Linton
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Isomorphism Class ◽  
Countably Infinite ◽  
Natural Way

AbstractFor countable structures and , let abbreviate the statement that every sentence true in also holds in . One can define a back and forth game between the structures and that determines whether . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω sentence, then there are countably infinite models and such that ⊨ θ, ⊨ ¬θ, and . For countable languages ℒ there is a natural way to view ℒ structúres with universe ω as a topological space, Xℒ. Let [] = { ∊ Xℒ∣ ≅ } denote the isomorphism class of . Let and be countably infinite nonisomorphic ℒ structures, and let C ⊆ ωω be any subset. Our main result states that if , then there is a continuous function f: ωω → Xℒ with the property that x ∊ C ⇒ f(x) ∊ [] and x ∉ C ⇒ f(x) ∊ f(x) ∈ []. In fact, for α ≤ 3, the continuous function f can be defined from the relation.

THE FUNDAMENTAL GROUP OF THE HAWAIIAN EARRING IS NOT FREE

1992 ◽  
Vol 02 (01) ◽  
pp. 33-37 ◽  
Author(s):  
BART DE SMIT
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Free Group ◽  
Arbitrary Number ◽  
Short Proof ◽  
Single Line ◽  
Countably Infinite ◽  
Hawaiian Earring

The Hawaiian earring is a topological space which is a countably infinite union of circles, that are all tangent to a single line at the same point, and whose radii tend to zero. In this note a short proof is given of a result of J.W. Morgan and I. Morrison that describes the fundamental group of this space. It is also shown that this fundamental group is not a free group, unlike the fundamental group of a wedge of an arbitrary number of circles.

scholarly journals Ext(Q,Z) is the additive group of real numbers

1969 ◽  
Vol 1 (3) ◽  
pp. 341-343 ◽  
Author(s):  
James Wiegold
Keyword(s):  
Cyclic Group ◽  
Additive Group ◽  
Rational Numbers ◽  
Infinite Subset ◽  
Real Numbers ◽  
Infinite Cyclic Group ◽  
Extra Effort ◽  
Uncountable Group ◽  
Countably Infinite

Standard homological methods and a theorem of Harrison on cotorsion groups are used to prove the result mentioned.In this note Z denotes an infinite cyclic group, Q the additive group of rational numbers, Zp ∞ a p–quasicyclie group, and Ip the group of p–adic integers.Pascual Llorente proves in [3] that Ext(Q,z) is an uncountable group, and gives explicitly a countably infinite subset. Very little extra effort produces the result embodied in the title, as follows.

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

Productly linearly independent sequences

2015 ◽  
Vol 52 (3) ◽  
pp. 350-370
Author(s):  
Jaroslav Hančl ◽  
Katarína Korčeková ◽  
Lukáš Novotný
Keyword(s):  
Rational Numbers ◽  
Infinite Products ◽  
Linearly Independent ◽  
New Concepts ◽  
Infinite Sequences

We introduce the two new concepts, productly linearly independent sequences and productly irrational sequences. Then we prove a criterion for which certain infinite sequences of rational numbers are productly linearly independent. As a consequence we obtain a criterion for the irrationality of infinite products and a criterion for a sequence to be productly irrational.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

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