Open Sets

Stand at any point of the open interval (0,1) — say x = 0.5, or x = 0.999 — and ask a simple question: can I take a tiny step in either direction and still be inside the set? For (0,1) the answer is always yes. Even sitting at 0.999 you still have a sliver of room to the right before you hit 1. That "everyone has a little breathing room" property is the whole idea of an open set — and it turns out to be one of the most foundational ideas in all of higher mathematics.

An open set has no boundary skin of its own: no point is pinned right against an edge with nowhere to go. From every member you can wiggle a bit and stay inside. Contrast the closed interval [0,1]: the endpoint 1 is a member, but every step to its right immediately falls out of the set. That single cornered point is enough to make [0,1] not open.

The definition

We make "breathing room" precise with a radius. A set U \subseteq \mathbb{R} is open if for every point x \in U there exists some radius \varepsilon > 0 — however small — such that the entire \varepsilon-ball around x lies inside U:

x \in U \;\Longrightarrow\; (x-\varepsilon,\, x+\varepsilon) \subseteq U \quad\text{for some } \varepsilon > 0.

The little interval (x-\varepsilon, x+\varepsilon) is called a neighbourhood of x. The definition says: every point of an open set is an interior point — it has a neighbourhood that fits wholly inside. The crucial phrase is "for some \varepsilon": the radius is allowed to depend on the point, and it may need to be tiny, but as long as a positive one exists at each point, the set is open.

The prototype is the open interval (a,b) — the round brackets mean the endpoints are left out — which is exactly why it is open. In higher dimensions the same idea uses round balls: an open disk in the plane (all points strictly inside a circle) is open, because any point inside has room for a small disk around it; the closed disk (which includes the rim) is not, because the rim points are cornered.

See the breathing room

The open interval (1,4) below has hollow endpoints — they are not members. Each interior point carries a little neighbourhood that fits entirely inside. Slide along and the room shrinks as you near an end, but for any point strictly inside it is still positive — that is exactly what "open" demands.

Worked examples

1. Why (0,1) is open. Take any x \in (0,1), so 0 < x < 1. Its distance to the nearer endpoint is d = \min(x,\, 1-x) > 0. Choose \varepsilon = d/2. Then (x-\varepsilon, x+\varepsilon) stays strictly between 0 and 1, so it lies inside (0,1). A valid \varepsilon exists at every point — the set is open. Notice the radius shrinks as x nears an end, but it never has to reach 0.

2. Why [0,1] is not open. Openness must hold at every point, so a single failure kills it. Look at the point x = 1. Any neighbourhood (1-\varepsilon, 1+\varepsilon) contains numbers bigger than 1 (for instance 1+\varepsilon/2), which are not in [0,1]. No positive radius works at 1, so [0,1] fails the definition.

3. The two edge cases. The whole line \mathbb{R} is open: every point trivially has room, since any neighbourhood is a subset of \mathbb{R}. The empty set \varnothing is open too — "every point of \varnothing has room" is vacuously true because there are no points to check. Both slip through the definition, and both matter as the anchors of the theory.

4. Open vs. closed disk in 2D. The open disk \{(x,y) : x^2 + y^2 < 1\} is open — a point at distance r < 1 from the centre has a small disk of radius (1-r)/2 entirely inside. The closed disk \{x^2 + y^2 \le 1\} is not: a point on the boundary circle has neighbours just outside, no matter how small the disk you draw.

Open, closed, or neither

The single most common confusion about open sets is that "open" and "closed" behave like a light switch — on or off, one or the other. They do not. A set is closed if its complement is open (equivalently, if it contains all of its limit/boundary points), and that is a separate condition. The four combinations all really occur:

No — and assuming it does is the classic trap. The half-open interval [0,1) is a live counterexample: it is not open (no room at 0) and also not closed (its complement (-\infty,0)\cup[1,\infty) is not open, because 1 is a boundary point it's missing). So "not open" tells you nothing about whether a set is closed — you must check closedness on its own. And "clopen" sets like \mathbb{R} and \varnothing show the two properties aren't even mutually exclusive. Open and closed are not opposites; they are two independent yes/no questions.

Why we care

Open sets are the raw material of measure theory on the line. Declaring exactly which sets are "events" starts by trusting the open sets and closing up from there — that is precisely how the Borel σ-algebra is built: the smallest σ-algebra that contains every open set. Rigorous probability — the foundation under everything from a coin-flip to the price of an option — rests on which sets we are allowed to call measurable, and that construction begins right here.

Because open sets are the atoms of topology. To build a topological space you don't specify distances at all — you simply declare which sets count as "open" (subject to a few rules: arbitrary unions and finite intersections of open sets stay open, and the whole space and \varnothing are open). Astonishingly, once you know the open sets, you can recover all of continuity, limits, and convergence without ever mentioning \varepsilon again. The clean punchline:

f \text{ is continuous} \iff f^{-1}(V) \text{ is open for every open } V.

A function is continuous exactly when the preimage of every open set is open — no \delta's in sight. That is why this humble "points have breathing room" idea underpins real analysis, probability, and the exotic curved geometries of modern physics alike. Topology is, quite literally, the study of open sets.