Property Suggestion: Angelic

[Moniker1998]

Property Suggestion

Call $A\subseteq X$ relatively countably compact if every sequence $x_n\in A$ has a convergent subnet (in $X$).
A space $X$ is called angelic if for every relatively countably compact $A\subseteq X$ we have

1. $A$ is relatively compact
2. If $x\in \overline{A}$ then there is a sequence $x_n\in A$ with $x_n\to x$.

Rationale

This property appears Banach space theory by Fabian et al., definition 3.53.
It's an important property in functional analysis.

Relatonship to other properties

Just some simple ones, I don't know any complicated ones:
Metrizable $\implies$ Angelic
Angelic + Compact $\implies$ Frechet-Urysohn
Frechet-Urysohn + Compact $\implies$ Angelic
Angelic + Countably compact $\implies$ Compact