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 can “inherit” a topology from the parent set . It is easy to verify that the set , as runs through , is a topology on . This prompts the following definition of a relative topology.
Relative Topology or Inherited Topology
Let be a topological space and be a nonempty subset, then is a topology on , called the topology induced by on or a relative topology on . The pair is called the subspace of . The topology is also called the inherited topology.
In other words, if is a topological space and is a non empty subset of , the collection consisting of those subsets of which are obtained by the intersections of the members of with is called the relative topology on . It is clear from the definition of the relative topology that each of its members is obtained by the intersection of some members of with . It should be noted that not every subset of is a subspace of . The subset of is a subspace of if and only if the topology of is the relative topology.
Let with topology and, using the definition of relative topology generated the topology on will be is a relative topology.
Let be a topological space and be the subset of . Then every open subset of is also open in , if and only if itself is open in . In other words, the subspace of a discrete topological space is also a discrete space.