We have already seen that sequences are “adequate” to detect limit points, continuous functions, and compact sets in metrizable spaces. There is a generalization of the notion of sequence, called a net, that will do the same thing for an arbitrary topological space. We give the relevant definitions here, and leave the proofs as exercises. Recall that a relation ≼ on a set A is called a partial order relation if the following conditions hold:
(1) α ≼ α for all α.
(2) If α ≼ β and β ≼ α, then α = β.
(3) If α ≼ β and β ≼ γ, then α<γ.
Now we make the following definition:
A directed set J is a set with a partial order ≼ such that for each pair α, β of elements of J, there exists an element γ of J having the property that α ≼ γ and β ≼ γ.
Show that the following are directed sets:
(a) Any simply ordered set, under the relation ≤.
(b) The collection of all subsets of a set S, partially ordered by inclusion (that is, A ≼ B if A ⊂ B).
(c) A collection of subsets of S that is closed under finite intersections, partially ordered by reverse inclusion (that is A ≼ B if A ⊃ B).
(d) The collection of all closed subsets of a space X, partially ordered by inclusion.