A quotient ordered space

1968 ◽  
Vol 64 (2) ◽  
pp. 317-322 ◽  
Author(s):  
S. D. McCartan
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Partial Orders ◽  
Projection Mapping ◽  
Quotient Spaces ◽  
The Family ◽  
Ordered Space ◽  
Image Position

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the propertya partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

scholarly journals A model of quotient spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 13-18
Author(s):  
Hawete Hattab
Keyword(s):  
Simple Model ◽  
Topological Space ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Finite Model ◽  
Orbit Spaces ◽  
Quotient Spaces

AbstractLet R be an open equivalence relation on a topological space E. We define on E a new equivalence relation ̃ℜ̅ by x̃ ̃ℜ̅y if the closure of the R-trajectory of x is equal to the closure of the R-trajectory of y. The quotient space E/̃ ̃ℜ̅ is called the trajectory class space. In this paper, we show that the space E/̃ ̃ℜ̅ is a simple model of the quotient space E/R. This model can provide a finite model. Some applications to orbit spaces of groups of homeomorphisms and leaf spaces are given.

scholarly journals Some fixed-point theorems on locally convex linear topological spaces

1975 ◽  
Vol 13 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Topological Spaces ◽  
The Family ◽  
Image Position ◽  
Commutative Family ◽  
Locally Convex

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the setcontains a point of M. Then the family F has a common fixed-point in M.

Partial Orders on the 2-Cell

1975 ◽  
Vol 18 (3) ◽  
pp. 411-416
Author(s):  
E. D. Tymchatyn
Keyword(s):  
Partial Order ◽  
Metric Space ◽  
Closed Subset ◽  
Cartesian Product ◽  
Partial Orders ◽  
Partial Ordering ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Ordered Space

A partially ordered space is an ordered pair (X, ≤) where X is a compact metric space and ≤ is a partial ordering on X such that ≤ is a closed subset of the Cartesian product X×X. ≤ is said to be a closed partial order on X.

A Note on Whitney Maps

1980 ◽  
Vol 23 (3) ◽  
pp. 373-374 ◽  
Author(s):  
L. E. Ward
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Internal Structure ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Fundamental Importance ◽  
Recent Book ◽  
Partially Ordered ◽  
Ordered Space

In his recent book [3] Nadler observes that the property of admitting a Whitney map is of fundamental importance in studying the internal structure of hyperspaces, especially their arc structure. Nadler presents three distinct methods of constructing a Whitney map on the hyperspace 2X of nonempty closed subsets of a continuum.A partially ordered space is a topological space X endowed with a partial order ≤ whose graph is a closed subset of X×X. It is well-known (see, for example, [2], page 167) that if X is a regular Hausdorff space then 2X is a partially ordered space with respect to inclusion.

scholarly journals WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS

2018 ◽  
Vol 83 (1) ◽  
pp. 392-394
Author(s):  
LINDA BROWN WESTRICK
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Turing Degrees ◽  
Image Position ◽  
Scott Set

AbstractLet ${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in ${\cal D}$ also holds in ${{\cal D}^{\cal S}}$, where ${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of ${\cal S}$.

Embedding Smooth Dendroids in Hyperspaces

1979 ◽  
Vol 31 (1) ◽  
pp. 130-138 ◽  
Author(s):  
J. Grispolakis ◽  
E. D. Tymchatyn
Keyword(s):  
Continuous Function ◽  
Partial Order ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Vietoris Topology ◽  
Compact Metric Space ◽  
Partially Ordered ◽  
Arcwise Connected ◽  
Ordered Space ◽  
Image Position

A continuum will be a connected, compact, metric space. By a mapping we mean a continuous function. By a partially ordered space X we mean a continuum X together with a partial order which is closed when regarded as a subset of X × X. We let 2x (resp. C(X)) denote the hyperspace of closed subsets (resp. subcontinua) of X with the Vietoris topology which coincides with the topology induced by the Hausdorff metric. The hyperspaces 2X and C(X) are arcwise connected metric continua (see [3, Theorem 2.7]). If A ⊂ X we let C(A) denote the subspace of subcontinua of X which lie in A.If X is a partially ordered space we define two functions L, M : X → 2X by setting for each x ∊ X

scholarly journals Semigroup structures for families of functions, I. Some Homomorphism Theorems

1967 ◽  
Vol 7 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Algebraic Structure ◽  
Topological Structure ◽  
Continuous Functions ◽  
The Family ◽  
Image Position ◽  
Natural Way

This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

scholarly journals THE NATURAL PARTIAL ORDER ON THE SEMIGROUP OF ALL TRANSFORMATIONS OF A SET THAT REFLECT AN EQUIVALENCE RELATION

2013 ◽  
Vol 88 (3) ◽  
pp. 359-368
Author(s):  
LEI SUN ◽  
XIANGJUN XIN
Keyword(s):  
Partial Order ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Natural Partial Order ◽  
Maximal Elements ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Image Position

AbstractLet ${ \mathcal{T} }_{X} $ be the full transformation semigroup on a set $X$ and $E$ be a nontrivial equivalence relation on $X$. Denote $$\begin{eqnarray*}{T}_{\exists } (X)= \{ f\in { \mathcal{T} }_{X} : \forall x, y\in X, (f(x), f(y))\in E\Rightarrow (x, y)\in E\} ,\end{eqnarray*}$$ so that ${T}_{\exists } (X)$ is a subsemigroup of ${ \mathcal{T} }_{X} $. In this paper, we endow ${T}_{\exists } (X)$ with the natural partial order and investigate when two elements are related, then find elements which are compatible. Also, we characterise the minimal and maximal elements.

Separation axioms for topological ordered spaces

1968 ◽  
Vol 64 (4) ◽  
pp. 965-973 ◽  
Author(s):  
S. D. McCartan
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Separation Axioms ◽  
Ordered Space

A topological ordered space (X, , <) is a set X endowed with both a topology and a partial order <, and is usually denoted by (X, ), it being understood that the symbol ≤ is used to denote all partial orders.

Ordering Uniform Completions of Partially Ordered Sets

1974 ◽  
Vol 26 (3) ◽  
pp. 644-664 ◽  
Author(s):  
R. H. Redfield
Keyword(s):  
Partial Order ◽  
Partially Ordered Sets ◽  
Partial Orders ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Ordered Space ◽  
Uniformly Continuous

Let (P, ) be a (nearly) uniform ordered space. Let (P, ) be the uniform completion of (P, ) at . Several partial orders for P are introduced and discussed. One of these orders provides an adjoint to the functor which embeds the category of uniformly complete uniform ordered spaces in the category of uniform ordered spaces, both categories with uniformly continuous order-preserving functions. When P is a join-semilattice with uniformly continuous join, two of these orders coalesce to the unique partial order with respect to which P is a join-semilattice, P is a join-subsemilattice of P, and the join on P is continuous.

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