Compactness is one of the most important concepts in analysis. Intuitively, a compact set is "small enough" that certain nice properties hold—like being able to extract convergent subsequences from any sequence.
A subset A ⊂ ℝ is compact if every open covering of A has a finite subcovering.
In ℝ (and ℝⁿ), this is equivalent to being closed and bounded. This is the famous Heine-Borel Theorem!
Contains all its accumulation points (complement is open)
Contained in some interval [−M, M] for M ∈ ℝ