If x0 is a real number, we can turn the set R − {x0} into a directed set by writing a ≤ b if and only if |a − x0| ≥ |b − x0|. We then say that the reals have been directed towards x0. This is not a partial order.
If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed set by writing U ≤ V if and only if U contains V.
For every U: U ≤ U; since U contains itself.
For every U,V,W: if U ≤ V and V ≤ W, then U ≤ W; since if U contains V and V contains W then U contains W.
For every U, V: there exists the set U ∩V such that U ≤ U ∩V and V ≤ U ∩V; since both U and V contain U ∩V.
In a poset P, every subset of the form {a| a in P, a ≤x}, where x is a fixed element from P, is directed.
Note that directed sets need not be antisymmetric and therefore in general are not partial orders. However, the term is also frequently used in the context of posets. In this setting, a subset A of a partially ordered set (P,≤) is called a directed subsetiff
for any two a and b in A, there exists a c in A with a ≤ c and b ≤ c (directedness),
where the order of the elements of A is inherited from P. For this reason, reflexivity and transitivity need not be required explicitly.
Directed subsets are most commonly used in domain theory, where one studies orders for which these sets are required to have a least upper bound. Thus, directed subsets provide a generalization of (converging) sequences in the setting of partial orders as well.
A UC Berkeley engineer, entranced by the uncanny ability of insects like dragonflies and bumblebees to peer in all directions at once, has for the first time created microscopic versions of their compound eyes in the laboratory.
Luke Lee, professor of bioengineering and leader of a research team he calls the Bio-Poets, is reporting today that his synthetic devices made of complex plastic materials can "see" in all directions simultaneously, and could well find uses in fields as varied as medicine, 3-D cameras and even espionage.
The bulbous compound eyes of many insects contain thousands of individual lenses, each of which sees in a single direction, but whose images are melded into a single, wide-angle view that allows the insect to survey its entire neighborhood at once.
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