We shall describe a method of constructing new topologies from the given ones. If  is a topological space and is any subset, there is a natural way in which Y can “inherit” a topology from the parent set X. It is easy to verify that the set, as runs through, is a topology on Y. This prompts the following definition of (relative topology).

Relative Topology or Inherited Topology:
            Let be a topological space and a nonempty subset, then is a topology on Y, called the topology induced by  on Y or relative topology on Y. The pair is called the subspace of X. The topology is also called the inherited topology.
            In other words, if is a topological space and Y is a non empty subset of X. The collection consisting of those subsets of Y which are obtained by the intersections of the members of  with Y is called the relative topology on Y. It is clear from the definition of the relative topology,, that each of its member is obtained by the intersection of some members of  with Y. It should be noted that not every subset Y of X is a subspace of X. The subset Y of X is a subspace of X if and only if the topology of Y is the relative topology.

Example:
            Let  with topology  and, using the definition of relative topology  generated the topology on Y will be  is a relative topology.

Remark:
            Let be a topological space and Y is the subset of X. Then every open subsets of Y is also open in X, if and only if, Y itself is open in X. In other words the subspace of a discrete topological space is also discrete space.