Stability of positive and monotone systems in a partially ordered space

2004 ◽  
Vol 56 (4) ◽  
pp. 560-576 ◽  
Author(s):  
A. G. Mazko
Keyword(s):  
Monotone Systems ◽  
Partially Ordered ◽  
Ordered Space

A partially ordered space which is not a priestley space

1980 ◽  
Vol 20 (1) ◽  
pp. 293-297 ◽  
Author(s):  
Albert Stralka
Keyword(s):  
Priestley Space ◽  
Partially Ordered ◽  
Ordered Space

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.

On the Non-Cutpoint Existence Theorem

1968 ◽  
Vol 11 (2) ◽  
pp. 213-216 ◽  
Author(s):  
L. E. Ward
Keyword(s):  
Partial Order ◽  
Existence Theorem ◽  
Hausdorff Space ◽  
Fundamental Result ◽  
Partially Ordered ◽  
Ordered Space ◽  
Partially Ordered Spaces ◽  
Connected Spaces

The theorem of the title asserts that every non-degenerate continuum (that is, every compact connected Hausdorff space containing more than one point) contains at least two non-cutpoints. This is a fundamental result in set - theoretic topology and several standard proofs, each varying from the others to some extent, have been published. (See, for example, [1], [4] and [5]). The author has presented a less standard proof in [3] where the non-cutpoint existence theorem was obtained as a corollary to a result on partially ordered spaces. In this note a refinement of that argument is offered which seems to the author to be the simplest proof extant. To facilitate its exposition, the notion of a weak partially ordered space is introduced and the cutpoint partial order of connected spaces is reviewed.

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.

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

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