Closed set
In topology, a closed set is a subset of a topological space whose complement is an open set.[1] Equivalently, a set is closed if it contains all its limit points. In first-countable spaces such as metric spaces, it is closed if the limit of every convergent sequence in the set belongs to the set.[2] Closed sets exhibit key properties that mirror those of open sets in complementary fashion: the empty set and the entire space are closed, arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed.[3] In metric spaces, common examples include closed balls, which consist of all points at a distance less than or equal to a fixed radius from a center, as well as the integers within the real numbers under the standard topology.[4] These concepts are foundational in general topology and analysis, enabling the study of continuity, compactness, and convergence without relying on specific metrics.[5] Sets that are both open and closed, termed clopen sets, arise in disconnected spaces and play a role in understanding topological connectedness.[6]Core Concepts
Definition
In topology, a subset C of a topological space X is defined as closed if its complement X \setminus C is open.[6] Equivalently, C is closed if it contains all of its limit points.[7] A point p \in X is a limit point of C if every open neighborhood of p intersects C at some point other than p itself.[7] This condition ensures that C encompasses all points that are "arbitrarily close" to it in the topological sense, without relying on distances. The definition assumes familiarity with the basic structure of topological spaces, where open sets form a collection closed under arbitrary unions and finite intersections, and neighborhoods are open sets containing a given point.[7] The concept of closed sets originated in the early 20th-century development of general topology, particularly through Felix Hausdorff's work, which generalized notions from metric spaces to abstract topological spaces.[8] In his 1914 book Grundzüge der Mengenlehre, Hausdorff introduced closed sets as foundational elements, defined axiomatically to preserve topological invariance and overcome the restrictions of metric-based approaches, such as those limited to Euclidean spaces.[8] This framework allowed for the study of continuity and convergence in broader settings.[8]Relation to Open Sets
In any topological space, the collection of closed sets forms precisely the family of complements of open sets, establishing a fundamental duality between the two concepts. Specifically, a subset C of the space X is closed if and only if its complement X \setminus C is open. This equivalence arises directly from the definition of a topological space, where the open sets satisfy the axioms of including the empty set and the whole space, being closed under arbitrary unions, and closed under finite intersections; the corresponding properties for closed sets—containing the empty set and whole space, closed under arbitrary intersections, and closed under finite unions—follow by taking complements.[9] This duality ensures that the topology is inherently closed under complements, meaning that the complement of any open set is closed and vice versa, which is a key structural axiom underpinning the theory of topological spaces.[9] A special case of this duality occurs with clopen sets, which are subsets that are simultaneously open and closed. The empty set \emptyset and the entire space X are always clopen in any topological space, as their complements are each other and both satisfy the openness axioms. In general, clopen sets represent partitions that respect the topology without boundaries in the open-closed sense.[9] In connected topological spaces, this duality takes on added significance: the only clopen sets are \emptyset and X itself. A space is connected if it cannot be expressed as the union of two nonempty disjoint open sets, which equivalently means it admits no nontrivial clopen subsets; any proper nonempty clopen set would disconnect the space by serving as both an open and closed partition.[10] This uniqueness highlights the role of connectedness in restricting the duality's manifestations. The open-closed duality also lays the groundwork for concepts like the interior and boundary of a set, where the interior is the largest open subset contained within it, and the boundary can be intuitively viewed as the difference between the closure (the smallest closed set containing it) and the interior, though these are explored further elsewhere. This relation reinforces the symmetric framework of topology, allowing proofs and properties to be dualized by complementation.[9]Properties
Set Operations
In topological spaces, closed sets exhibit specific algebraic properties under set operations, forming a family that is stable under certain unions and intersections. The collection of all closed sets in a topological space X is closed under arbitrary intersections and finite unions, meaning the result of such an operation remains closed. This stability arises from the duality between closed sets and open sets, where a set is closed if and only if its complement is open.[3] The intersection of any collection of closed sets, whether finite or infinite, is itself closed. To see this, suppose \{F_i : i \in I\} is an arbitrary family of closed subsets of X. Then the complements \{F_i^c : i \in I\} are open sets. The complement of the intersection is given by \left( \bigcap_{i \in I} F_i \right)^c = \bigcup_{i \in I} F_i^c, which is open as an arbitrary union of open sets. Therefore, \bigcap_{i \in I} F_i is closed.[3][11] In contrast, the union of finitely many closed sets is closed, but arbitrary (infinite) unions need not be. For a finite collection \{F_1, \dots, F_n\} of closed sets, the complements \{F_1^c, \dots, F_n^c\} are open, and the complement of the union is \left( \bigcup_{k=1}^n F_k \right)^c = \bigcap_{k=1}^n F_k^c, a finite intersection of open sets, which is open. Thus, \bigcup_{k=1}^n F_k is closed. However, in spaces such as the real numbers with the standard topology, an infinite union of closed sets may fail to be closed.[3][11] These properties distinguish closed sets from open sets, which are instead closed under arbitrary unions but only finite intersections.[3]Closure Operator
In a topological space (X, \tau), the closure of a subset A \subseteq X, denoted \mathrm{cl}(A) or \overline{A}, is defined as the intersection of all closed sets in X that contain A. This makes \mathrm{cl}(A) the smallest closed set containing A with respect to inclusion. Equivalently, \mathrm{cl}(A) = A \cup A', where A' is the set of all limit points of A. The closure operator \mathrm{cl} satisfies three fundamental properties: it is extensive, idempotent, and monotonic. Extensiveness: For any A \subseteq X, A \subseteq \mathrm{cl}(A).Proof: By definition, \mathrm{cl}(A) is the intersection of all closed sets containing A. Each such closed set contains A, so their intersection also contains A. Thus, A \subseteq \mathrm{cl}(A).[8] Monotonicity: If A \subseteq B \subseteq X, then \mathrm{cl}(A) \subseteq \mathrm{cl}(B).
Proof: The family of closed sets containing B is a subfamily of the family of closed sets containing A, since any closed set containing B also contains A. The intersection over a smaller family yields a larger or equal set, so \mathrm{cl}(B) \supseteq \mathrm{cl}(A).[8] Idempotence: For any A \subseteq X, \mathrm{cl}(\mathrm{cl}(A)) = \mathrm{cl}(A).
Proof: First, \mathrm{cl}(A) \subseteq \mathrm{cl}(\mathrm{cl}(A)) by extensiveness. For the reverse inclusion, note that \mathrm{cl}(A) is closed (as an intersection of closed sets) and contains A, so it is one of the closed sets in the intersection defining \mathrm{cl}(A). Thus, \mathrm{cl}(A) \supseteq \mathrm{cl}(\mathrm{cl}(A)), since \mathrm{cl}(\mathrm{cl}(A)) is the smallest closed set containing \mathrm{cl}(A), and \mathrm{cl}(A) is already closed and contains itself. Combining both directions gives equality.[8] A subset A \subseteq X is closed if and only if \mathrm{cl}(A) = A. If A is closed, then \mathrm{cl}(A) = A by the definition of closure as the smallest closed set containing A. Conversely, if \mathrm{cl}(A) = A, then A equals its closure, which is always closed as an intersection of closed sets, so A is closed. The closure operator in a topological space satisfies the Kuratowski closure axioms, which provide a complete axiomatic characterization. These four axioms, formulated by Kazimierz Kuratowski, are:
- \mathrm{cl}(\emptyset) = \emptyset (the empty set has empty closure).
- A \subseteq \mathrm{cl}(A) for all A \subseteq X (extensiveness).
- \mathrm{cl}(\mathrm{cl}(A)) = \mathrm{cl}(A) for all A \subseteq X (idempotence).
- \mathrm{cl}(A \cup B) = \mathrm{cl}(A) \cup \mathrm{cl}(B) for all A, B \subseteq X (additivity).
Any operator satisfying these axioms defines a unique topology on X via the closed sets as the fixed points of the operator (sets A with \mathrm{cl}(A) = A). Monotonicity follows as a theorem from axioms 2 and 4.