Closed Set

A closed set is a subset of a topological space whose complement is an open set. In metric spaces, this means that whenever a sequence of points in the set converges to a point of the surrounding space, that limit also belongs to the set. Closedness is fundamental to topology and mathematical analysis, but it depends on the topology and ambient space, not merely on the elements of the subset. (pi.math.cornell.edu)

Definition and basic properties

Let $X$ be a space with topology $\tau$. A subset $F\subseteq X$ is closed precisely when

Thus open and closed sets describe the same topological structure through complementary conditions. Both the empty set and $X$ itself are closed. Arbitrary intersections and finite unions of closed sets are closed:

These rules follow from the corresponding rules for open sets using De Morgan’s laws. Conversely, a family satisfying these closed-set rules determines a topology by taking complements. (pi.math.cornell.edu)

An infinite union of closed sets need not be closed. In the usual topology on the real numbers,

which excludes its limiting endpoint $0$. “Closed” is not synonymous with “not open”: some sets have both properties, while others have neither. For instance, $[0,1)$ is neither open nor closed in $\mathbb R$. (jirilebl.github.io)

Closure and neighborhoods

The closure of a subset $A$, denoted $\overline A$, is the intersection of all closed subsets containing $A$. It is therefore the smallest closed set containing $A$, and

A point belongs to $\overline A$ exactly when every open neighborhood of that point intersects $A$. Consequently, $F$ is closed if and only if every point outside $F$ has an open neighborhood disjoint from $F$. (jirka.org)

In a metric space $(X,d)$, the last condition can be expressed using an open ball: for every $x\notin F$, there is an $\varepsilon>0$ such that

This characterizes closedness by the ability to separate each exterior point locally from the set, without requiring a uniform separation distance for all exterior points. (jirka.org)

Sequences and examples

For a metric space, $F$ is closed if and only if every sequence $(x_n)$ in $F$ that converges in $X$ has its limit in $F$. To see the nontrivial direction, if $x\in\overline F$, choose $x_n\in F$ with $d(x_n,x)<1/n$. Then $x_n\to x$, so the assumed sequence condition forces $x\in F$. In arbitrary topological spaces, closed sets still contain all such limits, but the converse need not hold. (webpages.ciencias.ulisboa.pt)

Under the usual real topology, closed intervals $[a,b]$, rays such as $[a,\infty)$, and the set of integers are closed. Closedness does not imply boundedness: both $\mathbb Z$ and $\mathbb R$ are unbounded. The set

is not closed because it omits $0$, whereas adjoining $0$ makes it closed. The rational numbers are not closed in $\mathbb R$: their closure is all of $\mathbb R$, making them a dense subset. (jirilebl.github.io)

Dependence on the ambient space

If $Y\subseteq X$ carries the subspace topology, a subset $F\subseteq Y$ is closed in $Y$ exactly when

for some closed subset $C$ of $X$. Thus $(0,1]$ is not closed in $\mathbb R$, but is closed in $Y=(0,\infty)$, since it equals $Y\cap(-\infty,1]$. Every space is closed as a subset of itself. (jirilebl.github.io)

Changing the topology also changes closedness. In the topology induced by the discrete metric, every subset is both open and closed. Such examples show why specifying the topology is essential when discussing unfamiliar spaces. (jirka.org)

Continuity, compactness, and completeness

A function $f:X\to Y$ is continuous exactly when the inverse image $f^{-1}(C)$ of every closed subset $C\subseteq Y$ is closed in $X$. Images of closed sets need not be closed: for example, the continuous exponential function maps the closed set $\mathbb R$ onto the nonclosed subset $(0,\infty)$ of $\mathbb R$. (webpages.ciencias.ulisboa.pt)

Closedness and compactness are distinct. A closed subset of a compact space is compact, and a compact subset of a Hausdorff space is closed. In Euclidean space $\mathbb R^n$, the Heine–Borel theorem states that compact subsets are exactly the closed and bounded subsets. This equivalence fails for general metric spaces. (pi.math.cornell.edu)

A closed subset of a complete metric space is complete with the inherited metric: every Cauchy sequence converges in the ambient space, and closedness keeps its limit inside the subset. Conversely, a complete metric subspace is closed in its ambient metric space. Closedness alone does not imply completeness if the ambient space is incomplete. (jirka.org)