Closed Set

Complement of an Open Subset

This article delves into the concept of the complement of an open set, a fundamental notion in geometry, topology, and the broader landscape of mathematics. It's crucial to distinguish this from the idea of a set that is "closed" under a particular operation, a concept explored under closure (mathematics). Furthermore, for those seeking clarity on the term "closed" in other contexts, a Closed (disambiguation) page exists.

At its core, a closed set is defined by its relationship with open sets. In a topological space, a set is deemed closed if and only if its complement is an open set. This definition is not merely a matter of semantics; it carries significant implications. A different, yet equivalent, perspective in a topological space is that a closed set is one that meticulously contains all of its limit points. For those navigating the intricacies of a complete metric space, a closed set exhibits another property: it remains closed under the operation of taking limits. It's important not to conflate this with the notion of a closed manifold, which pertains to a different area of mathematical inquiry.

There exists a peculiar subset of sets within any topological space: those that are simultaneously open and closed. These are known as clopen sets, a category that often sparks curiosity due to its seemingly paradoxical nature.

Definition

Consider a topological space denoted as (X, τ), where X represents the underlying set and τ signifies the collection of open sets. Within this framework, the following statements are not just related, but entirely equivalent regarding a subset A of X (A ⊆ X):

A is a closed set in X. This is the primary definition we're exploring.
The complement of A, denoted as A^c (which is X \ A), is an open subset of (X, τ). This means A^c must be an element of the topology τ. This is the definitional link we are examining.
A is precisely equal to its own closure in X. The closure of a set, often denoted as cl(A) or Ā, is the smallest closed set containing A. If a set is equal to its closure, it implies it already contains all the points "added" to make it closed, meaning it must have been closed to begin with.
A contains all of its limit points. A limit point of a set is a point such that every neighborhood of it contains at least one point from the set, other than the point itself. If a set includes all such points, it satisfies the condition of being closed.
A contains all of its boundary points. The boundary of a set consists of points that are neither in its interior nor in the interior of its complement. If a set contains all its boundary points, it essentially "encloses" itself completely.

An alternative, and often quite useful, way to characterize closed sets is through the lens of sequences and nets. A subset A of a topological space X is closed in X if and only if every limit of every net whose elements are drawn from A also belongs to A. This means if a net of points within A converges, its limit must also reside within A. In spaces where sequences are sufficient to capture topological properties, specifically in first-countable spaces (which include metric spaces), it's enough to consider only convergent sequences instead of all nets. This characterization holds significant value as it can even serve as the very definition in the context of convergence spaces, which are more general structures than topological spaces. It's worth noting that this characterization inherently depends on the larger space X, as the convergence of a sequence or net is determined by the points present within X.

Consider a point x within X. This point x is said to be "close" to a subset A ⊆ X if x belongs to the closure of A in X (x ∈ cl_XA). This is equivalent to stating that x belongs to the closure of the set A ∪ {x} within the topological subspace A ∪ {x}, which is endowed with the subspace topology inherited from X. This concept of "closeness" allows for a more intuitive, almost vernacular, understanding of closed sets: a subset is closed if and only if it contains every point that is close to it.

In terms of net convergence, a point x ∈ X is close to a subset A if there exists a net (valued) in A that converges to x. This rephrases the idea of "closeness" in a more formal, sequential manner.

Now, let's consider a situation where X is a topological subspace of a larger topological space Y. In this scenario, Y is referred to as a topological super-space of X. It's possible for a point in Y \ X (a point in Y but not in X) to be close to A, even though it's not an element of X itself. This is precisely how a subset A ⊆ X can be closed within X but not closed within the broader context of Y.

If A ⊆ X and Y is any topological super-space of X, then A is always a subset (potentially a proper one) of cl_YA, the closure of A within Y. This holds true even if A is a closed subset of X (meaning A = cl_XA); it's still possible for A to be a proper subset of cl_YA. However, A is a closed subset of X if and only if A = X ∩ cl_YA for some (or, equivalently, for every) topological super-space Y of X. This establishes a crucial link between closure within a subspace and its relationship with closures in larger spaces.

The concept of closed sets also provides a powerful means to characterize continuous functions. A map f: X → Y is continuous if and only if for every subset A ⊆ X, the image of the closure of A in X is contained within the closure of the image of A in Y, i.e., f(cl_XA) ⊆ cl_Y(f(A)). This can be translated into more accessible language: f is continuous if and only if for every subset A ⊆ X, f maps points that are close to A to points that are close to f(A). Similarly, a function f is continuous at a specific point x ∈ X if and only if whenever x is close to a subset A ⊆ X, then f(x) is close to f(A). This highlights how continuity preserves the notion of "closeness" across spaces.

More About Closed Sets

The definition of a closed set, as presented, relies on the concept of open sets, which is a cornerstone of topological spaces. This definition extends gracefully to other spaces that possess topological structures, such as metric spaces, differentiable manifolds, uniform spaces, and gauge spaces.

It's a crucial point that whether a set is considered closed is not an intrinsic property of the set itself, but rather depends on the space in which it is embedded. However, there's a special class of spaces: compact Hausdorff spaces. These spaces are, in a sense, "absolutely closed." This means that if you embed a compact Hausdorff space D into an arbitrary Hausdorff space X, D will invariably be a closed subset of X. The surrounding space, in this case, becomes irrelevant to the closedness of D. The process of Stone–Čech compactification, which transforms a completely regular Hausdorff space into a compact Hausdorff space, can be understood as the act of adjoining the limits of certain nets that do not converge within the original space.

Furthermore, there are significant relationships between closed sets and compactness. Every closed subset of a compact space is itself compact. Conversely, every compact subspace of a Hausdorff space is necessarily closed. These interconnections are vital for understanding the structure of topological spaces.

Closed sets also offer a precise characterization of compactness. A topological space X is compact if and only if every collection of non-empty closed subsets of X whose intersection is empty admits a finite subcollection that also has an empty intersection. This is known as the finite intersection property for compact spaces.

The concept of disconnectedness is also illuminated by closed sets. A topological space X is disconnected if there exist two disjoint, non-empty, open subsets A and B of X whose union is X. In a related vein, a space is totally disconnected if it possesses an open basis composed entirely of closed sets.

Properties

The properties of closed sets are foundational to topology and exhibit a remarkable consistency:

A closed set invariably contains its own boundary. This means that if you are situated "outside" a closed set, any small movement in any direction will keep you outside the set. This principle holds even if the boundary itself is the empty set, a situation that can arise, for instance, in the metric space of rational numbers with respect to the set of numbers whose square is less than 2.
The intersection of any collection of closed sets, regardless of whether the collection is finite or infinite, is always a closed set. This property is incredibly powerful for constructing new closed sets.
The union of a finite number of closed sets is also a closed set. However, the union of an infinite number of closed sets is not necessarily closed.
The empty set is considered a closed set.
The entire space X is also considered a closed set.

These properties are so fundamental that they can be used to define a topology. If you are given a set X and a collection F of its subsets such that the properties listed above (intersection of any family is in F, finite unions are in F, the empty set and X are in F) hold, then there exists a unique topology τ on X such that the closed subsets of (X, τ) are precisely the elements of F. This demonstrates the deep axiomatic significance of closed sets.

The intersection property is particularly instrumental in defining the closure of a set A within a space X. The closure of A, denoted cl(A), is defined as the smallest closed subset of X that contains A. This smallest closed superset is constructed precisely by taking the intersection of all closed supersets of A.

Sets that can be expressed as the union of countably many closed sets are known as F_σ sets. It's important to remember that while these sets are formed from closed sets, they are not necessarily closed themselves.

Examples

To solidify understanding, let's examine some concrete examples:

The closed interval [a, b] on the real number line is a classic example of a closed set. The notation, using square brackets, signifies that the endpoints 'a' and 'b' are included in the set.
The unit interval [0, 1] is closed within the metric space of real numbers. However, consider the set of rational numbers within this interval, [0, 1] ∩ ℚ. This set is closed within the space of rational numbers, but it is not closed within the real numbers. This illustrates how the embedding space dictates closedness.
Sets that are neither open nor closed are also common. The half-open interval [0, 1) on the real numbers serves as a prime example. It includes 0 but excludes 1, making it neither fully open nor fully closed.
The ray [1, +∞) is a closed set. It includes the point 1 and extends infinitely in the positive direction, and its complement is an open interval.
The Cantor set presents an intriguing case. It is a closed set, yet it possesses the unusual property of being nowhere dense (meaning its interior is empty and it doesn't contain any intervals) and consists entirely of boundary points.
Singleton points, and by extension any finite set of points, are closed in T₁ spaces and Hausdorff spaces. This means that each individual point is its own closed set.
The set of integers, denoted ℤ, forms an infinite and unbounded closed set within the real numbers. While it's sparse, it contains all its limit points (which are none, in this case, as it's a discrete set within the reals).
For a function f: X → Y between topological spaces, f is continuous if and only if the preimages of closed sets in Y are closed in X. This is the dual condition to mapping open sets to open sets for continuity.