Connectedness

Connectedness in topology describes spaces that cannot be split into separate pieces, capturing the idea of being "whole."

Definition

A space \(X\) is connected if it cannot be written as the union of two disjoint, non-empty open sets.

Examples

The interval \([0, 1]\) is connected, but \((0, 1) \cup (2, 3)\) is not, as it has two separate parts.

Types

Path-connectedness is stronger: every pair of points can be joined by a continuous path (e.g., a circle).