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.

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

Example:

very 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 connected subspace if Y is a 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 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 Connected Space:

In a space, the following are equivalent:
Xis 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\}.