Definition and Properties of Open and Closed Sets

Introduction

In the study of topology and analysis, the concepts of open and closed sets form the foundation for understanding continuity, convergence, and the structure of topological spaces. A set is termed open if it is contained within a topology, meaning that for every point in the set, there exists a neighborhood entirely within the set. Conversely, a set is closed if its complement is open, or equivalently, if it contains all its limit points. These definitions are pivotal in characterizing the behavior of functions and the connectivity of spaces. The properties of open and closed sets, such as their closure under finite intersections and unions, and their relationship with continuous mappings, are essential for advanced mathematical reasoning.

Definitions

A topological space is a pair $(X, \tau)$, where $X$ is a non-empty set and $\tau$ is a topology on $X$, a collection of subsets of $X$ satisfying the following axioms:

  1. The empty set and $X$ itself are in $\tau$.
  2. Any finite intersection of elements of $\tau$ is also in $\tau$.
  3. Any union of elements of $\tau$ is also in $\tau$.

A subset $U \subseteq X$ is open if it belongs to $\tau$. A subset $C \subseteq X$ is closed if its complement $X \setminus C$ is open.

The distinction between open and closed sets is critical. While open sets are defined by their local neighborhoods, closed sets are defined by their global closure properties. In a topological space, the intersection of any number of open sets is open, and the union of any number of open sets is open, ensuring that open sets form a topology. Similarly, closed sets are closed under finite intersections and unions, and their complements are open.

Properties of Open and Closed Sets

  1. Closure Under Finite Intersections and Unions:
    The collection of open sets in a topological space is closed under finite intersections and unions. This property ensures that the topological structure is preserved under these operations. For example, in the standard topology on $\mathbb{R}$, the union of open intervals $(a, b)$ and $(c, d)$ is also open, and their intersection is open.

  2. Complementarity:
    A set is closed if and only if its complement is open. This duality is fundamental in topology. For instance, in the discrete topology on a set $X$, every subset is both open and closed, as every subset is its own complement.

  3. Limit Points and Closed Sets:
    A set is closed if it contains all its limit points. In a topological space, a limit point of a set $A$ is a point $x$ such that every neighborhood of $x$ intersects $A$. If $A$ is closed, then $x \in A$ for all such $x$. This property ensures that closed sets are "complete" in the sense that they do not "miss" their boundary points.

  4. Relationship with Continuous Functions:
    A function $f: X \to Y$ between topological spaces is continuous if the preimage of every open set in $Y$ is open in $X$. In particular, if $f$ is a closed map, then it preserves the closedness of sets under inverse images. This relationship is crucial in understanding the behavior of functions in topological spaces.

  5. Topological Indicators:
    Open and closed sets are often used to define topological invariants. For example, a space is T₀ (discrete) if all singletons are closed, and T₁ if all singletons are open. The study of such spaces