I have read two different usage for the expression "Heine-Borel property":
$1)$ Rudin: Functional Analysis (1.8 definition, page 9) (also in wikipedia):
$X$ has the Heine-Borel property if every closed and bounded subset of $X$ is compact. (where $X$ denotes a topological vector space)
$2)$ Folland: Real Analysis (0.25 Theorem, page 15):
Heine-Borel property means, that every open cover of $E$ has a finite subcover. (where $E$ denotes a subset of a metric space)
I know, the last one is how we define a compact space. My question is, which one is used in functional analysis, and what could be the point, that Folland used this notation in $2)$?