Open and Closed Sets in Topology

Introduction

In topology, open and closed sets form the foundational framework for understanding the structure of spaces. These concepts extend the intuitive ideas of open and closed intervals in the real numbers to a broad range of spaces, far beyond numerical settings.

Open Sets

An open set is defined by the property that, for any point within the set, there exists an "enclosure" of other points in the set around it, without including a definitive boundary. This concept is crucial in the formal definition of a topological space.

Definition

Given a topological space $(X, \mathcal{T})$, where $X$ is a set and $\mathcal{T}$ is a collection of subsets of $X$, $\mathcal{T}$ is a topology on $X$ if it satisfies the following conditions:

1. The empty set $\emptyset$ and the set $X$ itself are included in $\mathcal{T}$.
2. The union of any collection of sets in $\mathcal{T}$ is also in $\mathcal{T}$.
3. The intersection of any finite collection of sets in $\mathcal{T}$ belongs to $\mathcal{T}$.

The elements of $\mathcal{T}$ are referred to as open sets.

Analogy with Open Intervals

The idea of an open set in topology is analogous to an open interval $(a, b)$ in the real numbers, where $a < b$. Similar to how an open interval encompasses all points between $a$ and $b$ without the endpoints, an open set includes points without their "boundary."

Closed Sets

Conversely, closed sets include both the "interior" points and the "boundary" points of the set, complementary to the concept of open sets.

Definition

In a topological space $(X, \mathcal{T})$, a set is considered closed if its complement with respect to $X$ is an open set.

Analogy with Closed Intervals

The concept of a closed set generalizes the idea of a closed interval $[a, b]$ in the real numbers, which includes all points between $a$ and $b$, including the endpoints $a$ and $b$. Similarly, a closed set in a topological space includes its boundary.

Conclusion

The notions of open and closed sets are pivotal in topology, facilitating the study of continuous functions, convergence, and other key properties without necessarily relying on a specific notion of distance. They enable the exploration of the spatial and geometric properties of various spaces, extending well beyond the familiar confines of the real numbers.