# Connected Space

A topological space which cannot be written as the union of two non-empty disjoint open sets is said to be a connected space.

In other words, a space $$X$$ is connected if it is not the union of two non-empty disjoint open sets.

**Example:**

Every indiscrete space is connected.

Let $$X$$ be an indiscrete space, then $$X$$ is the only non-empty open set, so we cannot find the disconnection of $$X$$. Hence $$X$$ is connected.

**Connected Subspace**

A subspace $$Y$$ of a topological space is said to be a connected subspace if $$Y$$ is connected as a topological space in its own right.

**Example:**

The subset of a connected space may not be connected.

The set of real numbers $$\mathbb{R}$$ cannot be represented as the union of two disjoint non-empty sets, so $$\mathbb{R}$$ is a connected space.

Next suppose that $$A = \mathbb{Q} \cap \left] { – \infty ,\sqrt 2 } \right[$$ and $$B = \mathbb{Q} \cap \left] {\sqrt 2 ,\infty } \right[$$

Since $$A = \mathbb{Q} \cap \left] { – \infty ,\sqrt 2 } \right[$$ and $$B = \mathbb{Q} \cap \left] {\sqrt 2 ,\infty } \right[$$ are open subsets of $$\mathbb{R}$$, so $$A$$ and $$B$$ are open subsets of $$\mathbb{Q}$$. Also $$A \cap B = \phi $$ and $$A \cup B = \mathbb{Q}$$. This shows that $$\mathbb{Q}$$ is a disconnected subspace of $$\mathbb{R}$$.

**Theorems**

• A topological space is connected if and only if it cannot be represented as the union of two disjoint non-empty closed sets.

• An infinite set with co-finite topology is a connected space.

• Any continuous image of a connected space is connected.

• The range of a continuous real unction defined on a connected space is an interval.

• If $$\left\{ {A,B} \right\}$$ is a disconnection of $$X$$ and $$C$$ is a connected subspace of $$X$$, then $$B$$ is contained either in $$A$$ or in $$B$$.

**Characterization of a Connected Space**

In a space, the following are equivalent:

• $$X$$is connected.

• The only open and closed subsets of $$X$$ are $$\phi $$, $$X$$.

• There does not exist a continuous map $$f:X \to \left\{ {0,1} \right\}$$ from a space $$X$$ onto the discrete space $$\left\{ {0,1} \right\}$$.