View source for Relatively-compact set

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A subset $A$ of a [[topological space]] $X$ with the property that the [[Closure of a set|closure]] $\bar A$ of $A$ in $X$ is [[Compact space|compact]].

A subset $A$ of a metric space $X$ is relatively compact if and only if every sequence of points in $A$ has a cluster point in $X$.

A space is compact if it is relatively compact in itself.

An alternative definition is that $A$ is relatively compact in $X$ if and only if every open cover of $X$ contains a finite subcover of $A$.  This formulation is equivalent to requiring that the set $A$ be [[way below]] $X$ with respect to set inclusion and the directed set of open subsets of $X$.

====References====
* N. Bourbaki, "General Topology" Volume 4 Ch.5-10, Springer [1974] (2007) ISBN 3-540-34399-7 {{ZBL|1107.54002}}
* G. Gierz, Karl Heinrich Hofmann, K. Keimel, J.D. Lawson, M. Mislove, Dana S. Scott, "A compendium of continuous lattices" Springer (1980)  ISBN 3-540-10111-X  {{MR|0614752}}  {{ZBL|0452.06001}}