A set U is open if for every point x in U, there exists an open interval (a, b) containing x that lies entirely in U. A set F is closed if its complement is open, equivalently if it contains all its limit points. Open and closed sets form the foundation of topology on ℝ.
From your work with epsilon-N convergence, you already think carefully about what it means to be "close" to a point — every point within ε of a limit eventually gets captured by a sequence. Open and closed sets formalize this intuition by asking: for each point in a set, how much "breathing room" does it have within that set? This question turns out to encode a surprising amount of structure.
A set U is open if every point has an open interval around it that stays inside U. The open interval (0, 1) is the canonical example: pick any point x in (0, 1), and there is always a small interval (x − δ, x + δ) that fits inside (0, 1), as long as δ is small enough. The point 0 is not in (0, 1), and this is why: any interval around 0 reaches into negative numbers, which are outside the set. Open sets are the sets where no point sits at the edge — every point is strictly interior. A closed set, by contrast, contains all its limit points: if a sequence in the set converges, its limit is also in the set. The closed interval [0, 1] is closed because any sequence of points in [0, 1] that converges must converge to something in [0, 1].
The complementary definition is equally useful: a set is closed if and only if its complement is open. This duality means that working with closed sets is just as clean as working with open sets — you can always translate between them. But resist the temptation to think "open" and "closed" are opposites in the everyday sense. A set can be neither open nor closed: the half-open interval [0, 1) contains its left endpoint 0 (so it has a boundary point, failing openness) but not its right limit point 1 (so it fails closedness). More surprisingly, a set can be both open and closed — the empty set ∅ and the entire real line ℝ are "clopen," satisfying both definitions vacuously.
The equivalent characterization of closed sets via limit points is especially powerful for analysis. It says that closed sets are precisely the sets that are stable under limits: you cannot escape a closed set by converging. This is why theorems about sequences (continuous functions, convergence, Cauchy sequences) often conclude "the limit lies in F" — being closed guarantees it. The Heine-Borel theorem, which you will encounter next, adds compactness to this picture: a subset of ℝ is compact if and only if it is both closed and bounded, and compact sets have the strongest sequential stability properties of all.
Open and closed sets are not merely technical definitions — they are the language in which continuity, limits, and convergence are expressed most cleanly. When you study epsilon-delta continuity in real analysis, you will see that a function is continuous if and only if preimages of open sets are open (equivalently, preimages of closed sets are closed). This rephrasing lifts continuity from a local epsilon-delta condition into a global statement about set structure, and it generalizes directly to arbitrary topological spaces where there is no notion of distance at all.