ordered space

Definition. A set $X$ that is both a topological space and a poset is variously called a topological ordered space, ordered topological space, or simply an ordered space. Note that there is no compatibility conditions imposed on $X$. In other words, the topology $𝒯$ 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 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 partially ordered space.

One can construct an ordered space from a set with fewer structures.

1. 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 $a\in \overline{\{b\}}$. The relation so defined turns out to be a partial order on $X$, called the specialization order, making $X$ an ordered space.

2. 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.

3. A slightly more useful example is to take a poset $P$, and take

   $$ℒ(P):=\{P-\uparrow x\mid x\in P\},$$

   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 lower topology on $P$.

4. Dually, if we take

   $$𝒰(P):=\{P-\downarrow x\mid x\in P\},$$

   as the subbasis, we get the upper topology on $P$, denoted by $\nu (P)$.

5. In the lower topology $\omega (P)$ of $P$, if $y\in P-\uparrow x$, then either (strict inequality) or $x\parallel y$ (incomparable with $x$). If $x$ is an isolated element, then $P-\uparrow x=P-\{x\}$. This means that $\{x\}$ is a closed set. Similarly, $\{x\}$ is closed in the upper topology $\nu (P)$.

   If $x$ is the top element of $P$, then $\{x\}$ is a closed set in $\omega (P)$, since $P-\uparrow x=P-\{x\}$ is open. Similarly $\{x\}$ 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 $P-\uparrow x$ in a more familiar fashion: $(-\mathrm{\infty },x)$. Similarly, $P-\downarrow x$ may be written as $(x,\mathrm{\infty })$.

6. Things get more interesting when we take the common refinement of $\omega (P)$ and $\nu (P)$. What we end up with is called the interval topology of $P$.

   When $P$ is totally ordered, the interval topology on $P$ has

   $$ℐ(P):=\{(x,y)\mid x,y\in P\}$$

   as a subbasis, where $(x,y)$ denotes the open poset interval, consisting of elements $a\in P$ such that . 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 $ℝ$ is precisely the interval topology generated by the linear order on $ℝ$.

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 $X$:

• $X$ is said to be upper semiclosed if $\uparrow x$ is a closed set for every $x\in X$.

• Similarly, $X$ is lower semiclosed if $\downarrow x$ is closed in $X$.

• $X$ is semiclosed if it is both upper and lower semiclosed.

• If $\le$, as a subset of $X\times X$, is closed in the product topology, then $X$ is called a pospace.

Other structures, such as ordered topological vector spaces, topological lattices (http://planetmath.org/TopologicalLattice), and topological vector lattices (http://planetmath.org/TopologicalVectorLattice) are ordered spaces with algebraic structures satisfying certain additional compatibility conditions. Please click on the links for details.