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
-
$A$ is relatively compact
- 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
Property Suggestion
Call$A\subseteq X$ relatively countably compact if every sequence $x_n\in A$ has a convergent subnet (in $X$ ).$X$ is called angelic if for every relatively countably compact $A\subseteq X$ we have
A space
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:$\implies$ Angelic$\implies$ Frechet-Urysohn$\implies$ Angelic$\implies$ Compact
Metrizable
Angelic + Compact
Frechet-Urysohn + Compact
Angelic + Countably compact