The term connectedness represents the state of being linked or joined together. We can write various definitions for connectedness in different ways taking various contexts into account. However, the final meaning of connectedness does not change even if we define it in different ways. Let’s understand the concept of connectedness with respect to different mathematical chapters here in this article, along with notations and examples.

Learn about Connectivity in graph theory.

Connectedness in Topology

In mathematics, connectedness is a basic topological property of sets corresponding to the intuitive thought of having no breaks. A set is said to be not connected or disconnected if it can be divided into two parts so that a point of one of the parts can never be a limit point of the other part. The set is called connected if it cannot be divided (as defined above).

Whether or not a set stays connected even after removing some of its points is one of the predominant methods of categorising figures in topology. For instance, if we removed a point from an arc, any remaining points on either side of the gap or break will not be limit points of the other side, so the resulting set is disconnected. However, if we remove one point from a simple closed curve, for example, a circle or polygon, it remains connected. When we remove two points, it becomes disconnected.

Connectedness in Real Analysis

In real analysis, we come across the term connectedness when we deal with metric spaces. Thus, we can define connectedness as follows.

A set in A in Rⁿ is connected if it is not a subset of the disjoint union of two open sets, and these two sets intersect.

(or)

A set X is called disconnected if there exists a continuous function f: X → {0, 1} and is constant. If no such function exists, we can tell that X is connected.

(or)

Disconnection of a set E in a metric space (X, d) consists of two nonempty sets E1, E2 such that the disjoint union of E1, E2 is E, and each of these two is open relative to E. Also, a set is connected if it does not have any such disconnections.

Connectedness Example

Two open, disjoint intervals cannot cover the set [0, 2]; for example, the open sets (-1, 1) and (1, 2) do not cover [0, 2] since the point x = 1 is not in their union. Therefore, [0, 2] is connected.

However, the set {0, 2} can be covered by the union of (-1, 1) and (1, 3). In this case, {0, 2} is not connected.

Properties of Connectedness

Some of the important properties of the connectedness of a set are listed below.

• A subset of a topological space is said to be connected if it is connected in the subspace topology.
• The interval (0, 1) ⊂ R with its usual topology is connected.
• Intervals are the only connected subsets of R with the usual topology.
• The continuous image of a connected space is connected. That means if f : [a, b] → R is continuous, the image of f is connected.