Let {xi , i ∈ I } be a net where I is a directed set. For J ⊂ I, {xi , i ∈ J } will be called a strict subnet of {xi , i ∈ I } if J is cofinal in I , that is, for all i ∈ I, i ≤ j for some j ∈ J .
(a) Show that this implies J is a directed set with the ordering of I .
(b) Show that in [0, 1] with its usual topology there exists a net having no convergent strict subnet (in contrast to Theorems 2.2.5 and 2.3.1). Hint: Let W be a well-ordering of [0, 1]. Let I be the set of all y ∈ [0, 1] such that {t : tW y} is countable. Show that I is uncountable and well-ordered by W . Let xy := y for all y ∈ I . Show that {xy : y ∈ I } has no convergent strict subnet.
Compactness can be characterized in terms of convergent subnets (e.g. Kelley, 1955, Theorem 5.2), but only for nonstrict subnets; see also Kelley (1955, p. 70 and Problem 2.E).