# ordered space

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\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.
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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 $(x,\infty)$.
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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