In topology, the concept of "closedness" refers to a particular type of subset within a topological space. A subset of a topological space is defined as closed if it contains all its limit points. This means that if a sequence (or more generally, a net or filter) of points within the closed set converges to a point, that point must also be within the set. Closed sets are a fundamental aspect of topological study and have many important properties and implications. ### Definition and Basic Properties **Formal Definition:** A subset �A of a topological space �X is closed if the complement �∖�X∖A is an open set. In other words, a set is closed if it contains its boundary. **Characteristics of Closed Sets:** - **Complements**: As mentioned, the complement of a closed set in the space is open. This duality between open and closed sets is a key feature of topology. - **Closure Operation**: A set is closed if it is equal to its closure. The closure of a set �A, denoted by �‾A, is the smallest closed set containing �A and is defined as the intersection of all closed sets containing �A or alternatively as the set of all limit points of �A. - **Finite Intersections and Arbitrary Unions**: The intersection of any number of closed sets is closed, and the union of a finite number of closed sets is also closed. However, the union of an infinite number of closed sets may not be closed. ### Examples in Common Spaces 1. **In �R (Real Numbers)**: - The set [�,�][a,b] (including endpoints �a and �b) is closed in �R because its complement (−∞,�)∪(�,∞)(−∞,a)∪(b,∞) is open. - Singleton sets, such as {�}{x} where �x is any real number, are closed because they contain all their limit points (in this case, just �x itself). 2. **In Metric Spaces**: - In any metric space, a closed ball defined by �‾(�,�)={�∈�:�(�,�)≤�}B(x,r)={y∈X:d(x,y)≤r} is closed, where �d represents the distance function. ### Theoretical and Practical Implications - **Limits and Convergence**: Closed sets are intimately related to the concepts of limits and convergence. They are crucial in the definition and analysis of convergent sequences in topological spaces. - **Compactness**: In metric spaces, a subset is compact if and only if it is closed and bounded. Compactness is a highly desirable property in many areas of mathematics, as it generalizes the notion of finiteness and often allows the application of various powerful theorems. - **Function Continuity**: A function between two topological spaces is continuous if the preimage of every closed set is closed. This characterization of continuity is especially useful in more abstract settings where traditional �−�ϵ−δ definitions are less practical. ### Role in Mathematical Analysis and Beyond Closed sets play a crucial role across all branches of mathematics, including analysis, algebra, and applied mathematics. They are used to define and study continuity, solve differential equations, and model physical phenomena where boundary conditions and limits play a central role. In algebraic topology, the concept of closedness helps define and understand topological invariants that classify spaces up to homeomorphism. # Conclusion Understanding closed sets provides foundational insights into the behavior of mathematical structures under limits and helps bridge the gap between pure theoretical mathematics and its practical applications in science and engineering. Also see [[Openness]]. # References ```dataview Table title as Title, authors as Authors where contains(subject, "closedness") or contains(subject, "Compactness") or contains(subject, "Bounded") sort modified desc, authors, title ```