Open Sets

An open set on the real line is one with no edge of its own: from every point in the set you can take a small step in either direction and still be inside it. There is always a little breathing room around each member — never a point pinned right against the boundary.

Precisely, a set U \subseteq \mathbb{R} is open if for every x \in U there is a radius \varepsilon > 0 with the whole interval around x still inside:

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

The prototype is the open interval (a,b) — the round brackets mean the endpoints are left out. The closed interval [a,b] is not open: at the endpoint b every step to the right immediately leaves the set, so b has no room.

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.

U \subseteq \mathbb{R} is open iff every x \in U has some (x-\varepsilon, x+\varepsilon) \subseteq U.
Any union of open sets is open; any finite intersection of open sets is open.
An infinite intersection need not be: \bigcap_{n\ge 1}\left(-\tfrac1n, \tfrac1n\right) = \{0\}, which is not open.
Every open subset of \mathbb{R} is a countable union of disjoint open intervals.

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.