Open & Closed Sets

Interior points, closure, and why closed is not just not open.

The idea

Open and closed sets are the basic vocabulary of analysis: limits, continuity, compactness and connectedness are all phrased through them.

A point $x$ is interior to a set $S \subseteq \mathbb{R}$ when some interval $(x - \varepsilon, x + \varepsilon)$ around it lies entirely inside $S$. The set $S$ is open when every one of its points is interior. It is closed when its complement $\mathbb{R} \setminus S$ is open — equivalently, when $S$ contains every one of its limit points, the values approached arbitrarily closely by points of $S$. In the half-open interval $[a, b)$, an interior point $x$ passes the interior test, the included endpoint $a$ fails it, and the excluded endpoint $b$ is a limit point the set omits.

Despite the names, open and closed are not opposites, and closed does not mean "not open". They are two separate tests applied to the same set, and a set may pass both, pass exactly one, or fail both. A set that passes both is called clopen.

The closure $\overline{S}$ is $S$ together with all of its limit points. It is the smallest closed set containing $S$, and $S$ is closed exactly when $\overline{S} = S$.

Ways to work on it

• Walkthrough. Open via interior points, the open/closed distinction, and sets that are both open and closed.
• Practice. Classify a subset of the real line as open, closed, neither, or both.
• Hardest. Classify the rationals ℚ in the real line and find their closure.

Not sure where to start? Take the ten-question placement test.