By L. Smith
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Additional info for Algebraic Topology, Gottingen 1984
Later on, we shall consider the following fundamental concepts regarding the strong neighborhoods which generate the strong topology of a PN space. 1. Let (V, ν, τ, τ ∗ ) be a PN space, then (i) A sequence (pn ) in V is said to be strongly convergent to a point p in V , and we write pn → p or lim pn = p, if for each λ > 0, there exists a positive integer m such that pn ∈ Np (λ), for n ≥ m; (ii) A sequence (pn ) in V is called a strong Cauchy sequence if for every λ > 0 there is a positive integer N such that νpn −pm (λ) > 1 − λ, whenever m, n > N ; (iii) The PN space (V, ν, τ, τ ∗ ) is said to be distributionally compact (Dcompact) if every sequence (pn ) in V has a convergent subsequence (pnk ).
Let νp := ε0 and assume, if possible, that p = θ; therefore, for every t > 0, one has F ( pt ) = 1. Since p > 0, this would imply F = ε0 , contrary to the assumption. This proves (N1). Property (N2) is obvious. 7 in Schweizer and Sklar (1983)). f. G. Since [−1] νp = p G[−1] , one has, for all p and q in V [τM (νp , νq )][−1] = νp[−1] + νq[−1] = p G[−1] + q G[−1] = ( p + q )G[−1] ≥ p + q G[−1] = νp[−1] so that νp+q ≥ τM (νp , νq ), viz. property (N3) holds. 8 in Schweizer and Sklar (1983)). Thus the argument just used yields, for every α ∈ [0, 1], [τM ∗ (ναp , ν(1−α)p )][−1] = [τM (ναp , ν(1−α)p ][−1] [−1] = α p G[−1] + (1 − α) p G[−1] = p G[−1] = νp+q , Hence the assertion.
2. For a pair (V, ν) that satisfies conditions (N 1) and (N 2) the following statements are equivalent: ˇ (a) (V, ν) satisfies also condition (S); (b) for all p ∈ V and for all α ∈ [0, 1] νp = τM (ναp , ν(1−α)p ). 5) Proof. For every F ∈ ∆+ let F [−1] denote the left continuous quasiinverse1 of F . 11), for all F, G, and H in ∆+ H = τM (F, G) if, and only if, H [−1] = F [−1] + G[−1] . 5) holds if, and only if, for all p ∈ V and all α ∈ [0, 1], [−1] [−1] + ν(1−α)p . 6) (a) ⇒ (b) For all p ∈ V, for every α ∈ [0, 1], and for every t ∈ R+ ναp = νp t α and ν(1−α)p (t) = νp t .