Definition · applies to subsets of a space

Elementary (subsets of $\reals$)

For all real numbers $a$ and $b$ such that $a < b$, the interval of numbers between $a$ and $b$ including $a$ and $b$ is called an closed interval, and is written $[a, b]$.

High school (subsets of $\reals^n$)

Consider a set $F$ of points on the number line, in the plane, or in space. The set $F$ is closed if $F$ contains all of its boundary points.

Undergraduate (metric spaces)

Consider a set $F$ in a metric space. The set $F$ is closed if $F$ contains the limit of every convergent sequence of points of $F$.

Graduate (topological spaces)

Consider a set $X$ with a topology. A set $F \subseteq X$ is closed if its complement $X \setminus F$ is open.

Examples

Examples

  • $[0, 1]$ High school
    The interval $[0, 1]$ is closed in $\reals$, because it contains its boundary points $0$ and $1$.
  • The empty set $\emptyset$ (definition) High school
    The empty set is closed in $\reals$, because its complement, $\reals$, is open.
  • The set of real numbers, $\reals$ High school
    The set $\reals$ is closed as a subset of itself, because its complement, $\emptyset$, is open.
  • The Cantor set Undergraduate
    The Cantor set is closed, because it is the intersection of the sets $C_0 = [0, 1]$, $C_1 = [0, \frac{1}{3}] \cup [\frac{2}{3}, 1]$, $C_2$, $\dots$ that remain after each step of the construction. Each $C_n$ is a union of finitely many closed intervals, so each $C_n$ is closed, and every intersection of closed sets is closed.

Non-examples

Things that may be mistaken for examples.

  • $[0, 1)$, as a subset of $\reals$ High school
    The interval $[0, 1)$ is not closed in $\reals$, because it does not contain its boundary point $1$.
  • $(0, 1)$ High school
    The interval $(0, 1)$ is not closed in $\reals$, because the sequence $1/n$ in $(0, 1)$ converges to $0$, which is not in $(0, 1)$.

Properties

  • Every intersection of closed sets is closed, and every union of finitely many closed sets is closed. A union of infinitely many closed sets need not be closed: the union of the intervals $[1/n, 1]$ is $(0, 1]$.
  • In a metric space, the three definitions agree: a set contains its boundary points if and only if it contains the limits of its convergent sequences, if and only if its complement is open.
  • “Closed” is not the opposite of “open”: a set can be both (such as $\emptyset$ and $\reals$ in $\reals$) or neither (such as $[0, 1)$).

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