ordered spaceOrderedSpace2007-05-16 16:54:552007-06-01 21:14:45Definitionupper topologylower topologyinterval topologyupper semiclosedlower semiclosedsemiclosedpospacetotally ordered space\usepackage{amssymb,amscd}
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\newcommand{\down}{\downarrow\!\!}\textbf{Definition}. A set $X$ that is both a topological space and a poset is variously called a \emph{topological ordered space}, \emph{ordered topological space}, or simply an \emph{ordered space}. Note that there is no compatibility conditions imposed on $X$. In other words, the topology $\mathcal{T}$ and the partial ordering $\le$ on $X$ operate independently of one another.
If the partial order is a total order, then $X$ is called a \emph{totally ordered space}. In some literature, a totally ordered space is called an ordered space. In this entry, however, an ordered space is always a \emph{partially} ordered space.
One can construct an ordered space from a set with fewer structures.
\begin{enumerate}
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For example, any topological space is trivially an ordered space, with the partial order defined by $a\le b$ iff $a=b$. But this is not so interesting. A more interesting example is to take a $T_0$ space $X$, and define $a\le b$ iff $a\in \overline{\lbrace b\rbrace}$. The relation so defined turns out to be a partial order on $X$, called the specialization order, making $X$ an ordered space.
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On the other hand, given any poset $P$, we can arbitrarily assign a topology on it, making it an ordered space, so that every poset is trivially an ordered space. Again this is not very interesting.
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A slightly more useful example is to take a poset $P$, and take $$\mathcal{L}(P):=\lbrace P-\up x\mid x\in P\rbrace,$$ the family of all set complements of principal upper sets of $P$, as the subbasis for the topology $\omega(P)$ of $P$. The topology $\omega(P)$ so generated is called the \emph{lower topology} on $P$.
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Dually, if we take $$\mathcal{U}(P):=\lbrace P-\down x\mid x\in P\rbrace,$$ as the subbasis, we get the \emph{upper topology} on $P$, denoted by $\nu(P)$.
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In the lower topology $\omega(P)$ of $P$, if $y\in P-\up x$, then either $y< x$ (strict inequality) or $x\shortparallel y$ (incomparable with $x$). If $x$ is an isolated element, then $P-\up x=P-\lbrace x\rbrace$. This means that $\lbrace x\rbrace$ is a closed set. Similarly, $\lbrace x\rbrace$ is closed in the upper topology $\nu(P)$.
If $x$ is the top element of $P$, then $\lbrace x\rbrace$ is a closed set in $\omega(P)$, since $P-\up x=P-\lbrace x\rbrace$ is open. Similarly $\lbrace x\rbrace$ is closed in $\nu(P)$ if $x$ is the bottom element in $P$.
If $P$ is totally ordered, there are no isolated elements. As a result, we may write $P-\up x$ in a more familiar fashion: $(-\infty,x)$. Similarly, $P-\down x$ may be written as $(x,\infty)$.
\item
Things get more interesting when we take the common refinement of $\omega(P)$ and $\nu(P)$. What we end up with is called the \emph{interval topology} of $P$.
When $P$ is totally ordered, the interval topology on $P$ has $$\mathcal{I}(P):=\lbrace (x,y)\mid x,y\in P\rbrace$$
as a subbasis, where $(x,y)$ denotes the \emph{open} poset interval, consisting of elements $a\in P$ such that $x<a<y$. Since finite intersections of open poset intervals is a poset interval, an open set in $P$ can be written as an (arbitrary) union of open poset intervals.
As an example, the usual topology on $\mathbb{R}$ is precisely the interval topology generated by the linear order on $\mathbb{R}$.
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\textbf{Remark}. It is a common practice in mathematics to impose special compatibility conditions on a structure having two inherent substructures so the substructures inter-relate, so that one can derive more interesting fruitful results. This is true also in the case of an ordered space. Let $X$ be an ordered space. Below are some of the common conditions that can be imposed on $X$:
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$X$ is said to be \emph{upper semiclosed} if $\up x$ is a closed set for every $x\in X$.
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Similarly, $X$ is \emph{lower semiclosed} if $\down x$ is closed in $X$.
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$X$ is \emph{semiclosed} if it is both upper and lower semiclosed.
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If $\le$, as a subset of $X\times X$, is closed in the product topology, then $X$ is called a \emph{pospace}.
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Other structures, such as ordered topological vector spaces, \PMlinkname{topological lattices}{TopologicalLattice}, and \PMlinkname{topological vector lattices}{TopologicalVectorLattice} are ordered spaces with algebraic structures satisfying certain additional compatibility conditions. Please click on the links for details.
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\bibitem{ghklms} G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, {\em Continuous Lattices and Domains}, Cambridge University Press, Cambridge (2003).
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