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Closed Set — Definition, Formula & Examples

A closed set is a set that contains all of its limit points — meaning if a sequence of points in the set converges, the limit itself also belongs to the set. Equivalently, a set is closed when its complement is open.

A subset SS of a topological space XX is closed if X∖SX \setminus S is an open set in XX. In a metric space, this is equivalent to requiring that for every convergent sequence {xn}⊆S\{x_n\} \subseteq S with lim⁡n→∞xn=L\lim_{n \to \infty} x_n = L, the limit L∈SL \in S.

How It Works

To determine whether a set is closed, you can use either of two equivalent approaches. First, check whether the complement of the set is open — that is, whether every point in the complement has a neighborhood entirely contained in the complement. Alternatively, examine whether the set contains all its limit points by testing whether any convergent sequence drawn from the set has its limit inside the set. In R\mathbb{R} with the standard topology, closed intervals [a,b][a, b], single points {c}\{c\}, and R\mathbb{R} itself are all closed sets. A set can be both open and closed (called "clopen"), or neither.

Worked Example

Problem: Determine whether the set S=[0,1]S = [0, 1] is closed in R\mathbb{R} with the standard topology.
Step 1: Compute the complement of SS in R\mathbb{R}.
R∖S=(−∞,0)∪(1,∞)\mathbb{R} \setminus S = (-\infty, 0) \cup (1, \infty)
Step 2: Check whether the complement is open. Both (−∞,0)(-\infty, 0) and (1,∞)(1, \infty) are open intervals, and the union of open sets is open.
Step 3: Since the complement is open, SS is closed by definition. We can verify: any convergent sequence in [0,1][0,1] has its limit in [0,1][0,1], confirming the set contains all its limit points.
Answer: S=[0,1]S = [0, 1] is a closed set in R\mathbb{R}.

Why It Matters

Closed sets are essential in real analysis for guaranteeing that limits stay within a set, which underpins results like the extreme value theorem: a continuous function on a closed and bounded subset of Rn\mathbb{R}^n attains its maximum and minimum. In topology, closed sets are dual to open sets and form the basis for defining continuity, compactness, and convergence in abstract spaces.

Common Mistakes

Mistake: Assuming a set that is not open must be closed.
Correction: Sets can be neither open nor closed. For example, [0,1)[0, 1) in R\mathbb{R} is not open (no neighborhood of 00 lies entirely inside it) and not closed (the limit point 11 is missing). Always verify the definition directly.