General Topology

10 Partially ordered topological spaces

10.1 Partially ordered topological spaces

A partially ordered topological space is a triple $\mathrm{(}\mathit{X}\mathrm{,}\mathcal{T}\mathrm{,}\le \mathrm{)}$ where X is a nonempty set, $\mathcal{T}$ is a topology on X, and ≤ is a partial order on X. A partial order provides a means to quantify preference, with $\mathit{a}\le \mathit{b}$ meaning b is preferable to a. Thus, partially ordered topological spaces are useful in situations where nearness and preference are required.

Notice than any topological space $\mathrm{(}\mathit{X}\mathrm{,}\mathcal{T}\mathrm{)}$ can be viewed as a partially ordered topological space $\mathrm{(}\mathit{X}\mathrm{,}\mathcal{T}\mathrm{,}=\mathrm{)}$ ordered by the trivial order or discrete order of equality. Thus, topological spaces may be viewed as partially ordered topological spaces having the trivial order.

The next example shows that in general, the topology ...