Usual Topology on Real

Usual Topology on \mathbb{R}:

A collection of subsets of \mathbb{R} which can be can be expressed as union of open intervals, forms a topology on \mathbb{R}, and is called topology on \mathbb{R}.

Remark:

Every open interval is an open set but the converse may not be true.

 A = \left\{ {x \in \mathbb{R}:2 < x < 3{\text{ or }}4 < x < 5} \right\}


Usual Topology on {\mathbb{R}^2}:

Consider the Cartesian plane {\mathbb{R}^2}, then the collection of subsets of {\mathbb{R}^2} which can be expressed as union of open discs or open rectangles with edges parallel to coordinate axis from a topology, and is called usual topology on {\mathbb{R}^2}.

Usual Topology on {\mathbb{R}^3}:

Consider the Cartesian plane {\mathbb{R}^3}, then the collection of subsets of {\mathbb{R}^3} which can be expressed as union of open spheres or open cubed with edges parallel to coordinate axis from a topology, and is called usual topology on {\mathbb{R}^3}.