1. Let S be any set and S∞ the set of all sequences {xn }n≥1 with xn ∈ S for all n. Let C be a subset of the Cartesian product S × S∞. Also, S × S∞ is the set of all sequences {xn }n≥0 with xn ∈ S for all n = 0, 1,... . Such a set C will be viewed as defining a sense of "convergence," so that xn →C x0 will be written in place of {xn }n≥0 ∈ C . Here are some axioms: C will be called an L-convergence if it satisfies (1) to (3) below.
(1) If xn = x for all n, then xn →C x .
(2) If xn →C x , then any subsequence xn(k) →C x .
(3) If xn →C x and xn →C y, then x = y.
If C also satisfies (4), it is called an L∗-convergence:
(4) If for every subsequence k f→ xn(k) there is a further subsequence j f→ y j := xn(k( j )) with y j →C x , then xn →C x.
(a) Prove that if T is a Hausdorff topology and C (T ) is convergence for T , then C (T ) is an L∗-convergence.
(b) Let C be any L-convergence. Let U ∈ T (C ) iff whenever xn →C x and x ∈ U , there is an m such that xn ∈ U for all n ≥ m. Prove that T (C ) is a topology.
(c) Let X be the set of all sequences {xn }n≥0 of real numbers such that for some m, xn = 0 for all n ≥ m. If y(m) = {y(m)n }n≥0 ∈ X for all m = 0, 1,... , say y(m) →C y(0) if for some k, y(m) j = y(0) j = 0 for all j ≥ k and all m, and y(m)n → y(0)n as m → ∞ for all n. Prove that →C is an L∗-convergence but that there is no metric e such that y(m) →C y(0) is equivalent to e(y(m), y(0)) → 0.