# 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 $\left\{ {0,1} \right\}$.