Closed set

In topology, a closed set is a set which contains all of its limit points. Equivalently, a set in some topological space (including, for example, any metric space) is closed if and only if its complement is an open set, or alternatively if its closure is equal to itself.

In any topological space, the empty set and the entire space are both closed sets. The intersection of any collection of closed sets is a closed set. Also, the union of any two closed sets (or any finite number of closed sets) is a closed set. Note, however, that the union of an arbitrary collection of closed sets does not have to be closed. For example, every one-point set is closed in $\mathbb{R}$ (or in any $T_1$ space), but clearly the union of any arbitrary number of one-point sets does not need to be closed.

One common example of a closed set is the closed interval $[a, b] = \{x \mid a \leq x \leq b\}$ of the real numbers. However, closed subsets of $\mathbb{R}$ can take a variety of more complicated forms. For example, the set $\left\{\frac{1}{n} \mid n \in \mathbb{Z}_{> 0} \right\} \cup \{0\}$ is closed, as is the Cantor set.

The pre-image of a closed set under a continuous function is also closed.

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