A space is compact if every open cover has a finite subcover. This topological definition generalizes the Heine-Borel property from ℝ to arbitrary spaces without requiring a metric. Compactness is a fundamental property ensuring solutions exist for optimization problems and functions behave nicely.
From your study of open sets, you know that a topology on a space X is a collection of "open" subsets satisfying certain axioms — closed under arbitrary unions and finite intersections. An open cover of X is a collection {Uα} of open sets whose union contains all of X: X ⊆ ∪Uα. The definition of compactness asks: can you always get away with finitely many of them? A space is compact if for every open cover, no matter how the cover is constructed, you can select a finite subcollection that still covers X. The word "every" is doing enormous work here — you must be able to extract a finite subcover from any open cover, not just from nice ones.
To feel why this matters, consider the real line ℝ. Cover ℝ with the intervals (n−1, n+1) for every integer n. This open cover has no finite subcover — any finite subcollection only covers a bounded piece of ℝ. So ℝ is not compact. Now consider the closed interval [0,1]. It turns out that every open cover of [0,1] has a finite subcover — this is the content of the Heine-Borel theorem. The key features are that [0,1] is both closed (contains its limit points) and bounded (fits inside a ball of finite radius). In ℝⁿ, Heine-Borel says these two conditions are equivalent to compactness. In general topological spaces — where there is no notion of "bounded" — the open-cover definition is the right generalization.
The open-cover definition may feel abstract, but it captures an important geometric intuition: compact spaces are "small" in spirit even when they are not literally small. They cannot be "escaped" by sequences of points: in a compact space, every sequence has a convergent subsequence (this is sequential compactness, which is equivalent to compactness for metric spaces). This means optimization works: a continuous function on a compact space must attain its maximum and minimum values, because it cannot "escape to infinity" or approach a limit without reaching it. The Extreme Value Theorem from calculus is a special case — the domain [a,b] is compact.
To develop intuition for the definition, try to build an open cover of (0,1) with no finite subcover. One way: take Uₙ = (1/n, 1) for each positive integer n. This is an open cover of (0,1) — every point x ∈ (0,1) satisfies x > 1/n for large enough n, so x ∈ Uₙ. But any finite subcollection {U_{n₁}, ..., U_{nₖ}} only covers (1/N, 1) where N = max(n₁,...,nₖ), leaving the interval (0, 1/N] uncovered. The open interval (0,1) fails the compact definition because of the missing endpoint 0 — the "escape route" for the cover. The closed interval [0,1] plugs both endpoints and becomes compact. Compactness is, at its heart, the topological formalization of having no escape routes.