Connectedness Basics

Connectedness in topology describes spaces that cannot be split into separate pieces, capturing the idea of being "whole." This guide from MathMultiverse explores the definition, types, and examples of connectedness, providing a clear foundation for understanding this essential topological property.

Definition

A topological space \(X\) is connected if it cannot be written as the union of two disjoint, non-empty open sets. Formally, \(X\) is connected if there do not exist open sets \(U\) and \(V\) such that \(U \cap V = \emptyset\), \(U \cup V = X\), and both \(U\) and \(V\) are non-empty.

\[ X \text{ is connected} \iff \nexists \text{ open } U, V \text{ with } U \cap V = \emptyset, U \cup V = X, U \neq \emptyset, V \neq \emptyset \]

Examples

The interval \([0, 1]\) in \(\mathbb{R}\) is connected, as it cannot be split into two non-empty open sets without overlap. In contrast, the set \((0, 1) \cup (2, 3)\) is not connected, as it consists of two disjoint open intervals.

\[ (0, 1) \cup (2, 3) = \bigcup \{ (0, 1), (2, 3) \} \]

Types

Path-connectedness is a stronger condition than connectedness. A space is path-connected if any two points can be joined by a continuous path. For example, the unit circle in \(\mathbb{R}^2\) is path-connected, as any two points on the circle can be connected by an arc.

\[ \text{Path-connected: } \forall x, y \in X, \exists \gamma: [0, 1] \to X \text{ continuous with } \gamma(0) = x, \gamma(1) = y \]