For 1 ≤
p,
r <+∞,
f (≠ 0) ∈
Lp (
R,
dx), and
g (≠ 0) ∈
Lr (
R,
dx), the sequence space Λ
p (
f) with metric
dfp(
a,
b) was introduced in a previous paper and we discussed the inclusion relations between
lp and Λ
p (
f), and the linearity of Λ
p (
f).The purpose of this paper is to discuss the topological structures of (Λ
p (
f),
dfp).First we show that the space (Λ
p (
f),
dfp) is a complete separable metric group. Next we show that if Λ
p (
f) is a linear space, then (Λ
p (
f),
dfp) is a topological linear space. On the other hand, we give a necessary and sufficient condition for the inclusion Λ
p (
f) ⊂ Λ
r (
g). Furthermore, we show that the inclusions among the sequence spaces Λ
p (
f), Λ
r (
g) and
lr are continuous. The fact that Λ
p (
f) ⊂ Λ
r (
g) as sets implies the continuity of the inclusion (Λ
p (
f),
dfp) → (Λ
r (
g),
dgr) is emphasized.
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