Pseudometrizable Bornological Convergence is Attouch-Wets Convergence

Let

be an ideal of subsets of a metric space

\langle X,d \rangle ⟨ X , d ⟩

. A net of subsets

\langle A_\lambda\rangle ⟨ A λ ⟩

of

X X

is called

\mathcal{S} S

-convergent to a subset

A A

of

X X

if for each

S \in \mathcal{S} S ∈ S

and each

\varepsilon > 0 ε > 0

, we have eventually

A \cap S \subseteq A^\varepsilon_\lambda \ \textrm{and} \ A_\lambda \cap S \subseteq A^\varepsilon. A ∩ S ⊆ A λ ε  and  A λ ∩ S ⊆ A ε .

We identify necessary and sufficient conditions for this convergence to be admissible and topological on the power set of

X X

. We show that

\mathcal{S} S

-convergence is compatible with a pseudometrizable topology if and only if

\mathcal{S} S

has a countable base and each member of

\mathcal{S} S

has an

\varepsilon ε

-enlargement that is again in

\mathcal{S} S

. Further, in the case that the ideal is a bornology, we show that

\mathcal{S} S

-convergence when pseudometrizable is Attouch-Wets convergence with respect to an equivalent metric.