Closed set

Definition and Basic Properties

In topology, a branch of mathematics, a **closed set** is a set whose complement is an open set. More formally, a set \( C \) in a topological space \( (X, \tau) \) is closed if its complement \( X \setminus C \) is an element of the topology \( \tau \). This definition is fundamental to understanding various topological concepts and structures.

Closed sets have several important properties:

The union of a finite number of closed sets is closed.
The intersection of any number of closed sets is closed.
The entire space \( X \) and the empty set \( \emptyset \) are both closed sets.

These properties are dual to those of open sets, reflecting the complementary nature of the two concepts.

Examples of Closed Sets

Real Numbers

In the standard topology on the set of real numbers \( \mathbb{R} \), a set is closed if it contains all its limit points. For instance, the interval \([a, b]\) is closed because it includes its endpoints \( a \) and \( b \). Conversely, the interval \((a, b)\) is not closed since it does not include its endpoints.

Euclidean Space

In \( \mathbb{R}^n \) with the standard topology, closed sets include:

Closed intervals \([a, b]\) in \( \mathbb{R} \).
Closed balls \(\overline{B}(x, r)\) in \( \mathbb{R}^n \), defined as \(\{ y \in \mathbb{R}^n \mid \| y - x \| \leq r \}\).
The entire space \( \mathbb{R}^n \) and the empty set \( \emptyset \).

Metric Spaces

In a metric space \( (M, d) \), a set \( A \subseteq M \) is closed if it contains all its limit points. This means that if a sequence \((x_n)\) in \( A \) converges to a point \( x \in M \), then \( x \) must be in \( A \).

Closure and Interior

The **closure** of a set \( A \) in a topological space \( X \) is the smallest closed set containing \( A \). It can be constructed as the union of \( A \) and its limit points. The closure of \( A \) is often denoted by \( \overline{A} \).

The **interior** of a set \( A \), denoted by \( \text{int}(A) \), is the largest open set contained within \( A \). The interior of \( A \) consists of all points in \( A \) that are not on the boundary of \( A \).

Boundary and Accumulation Points

The **boundary** of a set \( A \), denoted by \( \partial A \), is the set of points that can be approached both from \( A \) and from its complement \( X \setminus A \). Formally, \( \partial A = \overline{A} \cap \overline{X \setminus A} \).

An **accumulation point** (or limit point) of a set \( A \) is a point \( x \) such that every neighborhood of \( x \) contains at least one point of \( A \) different from \( x \) itself. A set is closed if and only if it contains all its accumulation points.

Compactness and Closed Sets

A set is **compact** if every open cover has a finite subcover. In metric spaces, a set is compact if and only if it is closed and bounded. This is known as the Heine-Borel theorem. Compactness is a crucial concept in various areas of analysis and topology.

Closed Sets in Function Spaces

In the context of function spaces, closed sets play a significant role. For example, in the space of continuous functions, the set of functions that converge uniformly to a given function is closed. This property is essential in the study of functional analysis and operator theory.

Applications of Closed Sets

Closed sets are used extensively in various branches of mathematics:

In analysis, closed sets are used to define continuity, limits, and convergence.
In algebraic topology, closed sets are used to define and study various topological invariants.
In functional analysis, closed sets are crucial in the study of Banach and Hilbert spaces.