A glance into the anatomy of monotonic maps

<p>Given an autohomeomorphism on an ordered topological space or its subspace, we show that it is sometimes possible to introduce a new topology-compatible order on that space so that the same map is monotonic with respect to the new ordering. We note that the existence of such a re-ordering for a given map is equivalent to the map being conjugate (topologically equivalent) to a monotonic map on some homeomorphic ordered space. We observe that the latter cannot always be chosen to be order-isomorphic to the original space. Also, we identify other routes that may lead to similar affirmative statements for other classes of spaces and maps.</p>

The Distribution Function of a Probability Measure on a Linearly Ordered Topological Space

In this paper, we describe a theory of a cumulative distribution function on a space with an order from a probability measure defined in this space. This distribution function plays a similar role to that played in the classical case. Moreover, we define its pseudo-inverse and study its properties. Those properties will allow us to generate samples of a distribution and give us the chance to calculate integrals with respect to the related probability measure.

Antisymmetry of the Stochastical Order on all Ordered Topological Spaces

In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.

A NOTE ON PARACOMPACT p-SPACES AND THE MONOTONE D-PROPERTY

For any generalized ordered space X with the underlying linearly ordered topological space Xu, let X* be the minimal closed linearly ordered extension of X and $\tilde {X}$ be the minimal dense linearly ordered extension of X. The following results are obtained. (1)The projection mapping π:X*→X, π(〈x,i〉)=x, is closed.(2)The projection mapping $\phi : \tilde {X} \rightarrow X_u$, ϕ(〈x,i〉)=x, is closed.(3)X* is a monotone D-space if and only if X is a monotone D-space.(4)$\tilde {X}$ is a monotone D-space if and only if Xu is a monotone D-space.(5)For the Michael line M, $\tilde {M}$ is a paracompact p-space, but not continuously Urysohn.

Quasi-uniform completions of partially ordered spaces

In this paper we define partially ordered quasi-uniform spaces (X, $$\mathfrak{U}$$ , ≤) (PO-quasi-uniform spaces) as those space with a biconvex quasi-uniformity $$\mathfrak{U}$$ on the poset (X, ≤) and give a construction of a (transitive) biconvex compatible quasi-uniformity on a partially ordered topological space when its topology satisfies certain natural conditions. We also show that under certain conditions on the topology $$\tau _{\mathfrak{U}*} $$ of a PO-quasi-uniform space (X, $$\mathfrak{U}$$ , ≤), the bicompletion $$(\tilde X,\tilde {\mathfrak{U}})$$ of (X, $$\mathfrak{U}$$ ) is also a PO-quasi-uniform space ( $$(\tilde X,\tilde {\mathfrak{U}})$$ , ⪯) with a partial order ⪯ on $$\tilde X$$ that extends ≤ in a natural way.

Ordered compactifications with countable remainders

It is shown that if a partially-ordered topological space X admits a finite-point T2-ordered compactification, then it admits a countable T2-ordered compactification if and only if it admits n−point T2-ordered compactifications for all n beyond some integer m.

Posets of ordered compactifications

If (X′, τ′, ≤′) is an ordered compactification of the partially ordered topological space (X, τ, ≤) such that ≤′ is the smallest order that renders (X′, τ′, ≤′) a T2-ordered compactification of X, then X′ is called a Nachbin (or order-strict) compactification of (X, τ, ≤). If (X′, τ′, ≤*) is a finite-point ordered compactification of (X, τ, ≤), the Nachbin order ≤′ for (X′, τ′) is described in terms of (X, τ, ≤) and X′. When given the usual order relation between compactifications (ordered compactifications), posets of finite-point Nachbin compactifications are shown to have the same order structure as the poset of underlying topological compactifications. Though posets of arbitrary finite-point ordered compactifications are shown to be less well behaved, conditions for their good behavior are studied. These results are used to examine the lattice structure of the set of all ordered compactifications of the ordered topological space (X, τ, ≤).

On time dependent multistep dynamic processes

The discrete time scale Liapunov theory is extended to time dependent, higher order, nonlinear difference equations in a partially ordered topological space. The monotone convergence of the solution is examined and the speed of convergence is estimated.

Finite-point order compactifications

After the characterization of 1-point topological compactifications by Alexandroff in 1924, n-point topological compactifications by Magill [4] in 1965, and 1-point order compactifications by McCallion [5] in 1971, spaces that admit an n -point order compactification are characterized in Section 2. If X* and X** are finite-point order compactifications of X, sup{X*, X**} is given explicitly in terms of X* and X** in § 3. In § 4 it is shown that if an ordered topological space X has an m-point and an n-point order compactification, then X has a k-point order compactification for each integer k between m and n. The author is indebted to Professor Darrell C. Kent, who provided assistance and encouragement during the preparation of this paper.