Connectedness Examples

• \( X = \mathbb{R} \setminus \{0\} \) is disconnected, since \( (-\infty, 0) \) and \( (0, \infty) \) are disjoint open subsets of \( X \) whose union is \( X \).
• The space of rationals numbers \( \mathbb{Q} \), topologized as a subspace of \( \mathbb{R} \), is disconnected. Consider the pair of open sets \( (-\infty, \sqrt(2)) \cap \mathbb{Q} \) and \( (\sqrt(2), \infty) \cap \mathbb{Q} \).
• A discrete space with 2 or more points is disconnected. If \( X \) is such a space, we can choose an \( x_1 \in X \) then consider the two open sets \( \{x_1\} \) and \( X \setminus \{x_1\} \).
• A non-empty indiscrete space is connected.
• An interval topologized as a subspace of \( \mathbb{R} \) is connected.
• The space \( \mathbb{R}^d \) is not necessarily connected nor disconnected—it is connected in the product topology, but not connected in the box topology.
• The letter 'O' is connected. '0' is a quotient of \( [0,1] \), the quotient map is continuous and the image of a connected space through a continuous function is connected.
• The collection of connected sets in \( \mathbb{R} \) coincide preciesly with the collection of intervals.

Context