Connected Space - Examples

Examples

• The closed interval in the standard subspace topology is connected; although it can, for example, be written as the union of, the second set is not open in the aforementioned topology of .
• The union of is disconnected; both of these intervals are open in the standard topological space .
• (0, 1) ∪ {3} is disconnected.
• A convex set is connected; it is actually simply connected.
• A Euclidean plane excluding the origin, (0, 0), is connected, but is not simply connected. The three-dimensional Euclidean space without the origin is connected, and even simply connected. In contrast, the one-dimensional Euclidean space without the origin is not connected.
• A Euclidean plane with a straight line removed is not connected since it consists of two half-planes.
• The space of real numbers with the usual topology is connected.
• Any topological vector space over a connected field is connected.
• Every discrete topological space with at least two elements is disconnected, in fact such a space is totally disconnected. The simplest example is the discrete two-point space.
• On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring consists of two points and is connected. It is an example of a Sierpiński space.
• The Cantor set is totally disconnected; since the set contains uncountably many points, it has uncountably many components.
• If a space X is homotopy equivalent to a connected space, then X is itself connected.
• The topologist's sine curve is an example of a set that is connected but is neither path connected nor locally connected.
• The general linear group (that is, the group of n-by-n real matrices) consists of two connected components: the one with matrices of positive determinant and the other of negative determinant. In particular, it is not connected. In contrast, is connected. More generally, the set of invertible bounded operators on a (complex) Hilbert space is connected.
• The spectra of commutative local ring and integral domains are connected. More generally, the following are equivalent
  1. The spectrum of a commutative ring R is connected
  2. Every finitely generated projective module over R has constant rank.
  3. R has no idempotent (i.e., R is not a product of two rings in a nontrivial way).