Subspaces of Topology

We shall describe a method of constructing new topologies from the given ones. If \left( {X,\tau } \right) is a topological space and Y \subseteq X 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 V \cap Y, as V runs through \tau , is a topology on Y. This prompts the following definition of a relative topology.

Relative Topology or Inherited Topology

Let \left( {X,{\tau _X}} \right) be a topological space and Y \subseteq X be a nonempty subset, then {\tau _Y} = \left\{ {V \cap U:V \in {\tau _X}} \right\} is a topology on Y, called the topology induced by {\tau _X} on Y or a relative topology on Y. The pair \left( {Y,{\tau _Y}} \right) is called the subspace of X. The topology {\tau _Y} is also called the inherited topology.

In other words, if \left( {X,{\tau _X}} \right) is a topological space and Y is a non empty subset of X, the collection {\tau _Y} consisting of those subsets of Y which are obtained by the intersections of the members of {\tau _X} with Y is called the relative topology on Y. It is clear from the definition of the relative topology {\tau _Y} that each of its members is obtained by the intersection of some members of {\tau _X} 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 X = \left\{ {1,2,3,4} \right\} with topology {\tau _X} = \left\{ {\phi, \left\{ 2 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,2,3} \right\},X} \right\} andY = \left\{ {1,3,4} \right\} \subseteq X, using the definition of relative topology {\tau _Y} = \left\{ {V \cap U:V \in {\tau _X}} \right\} generated the topology on Y will be {\tau _Y} = \left\{ {\phi, \left\{ 1 \right\},\left\{ 3 \right\},\left\{ {1,3} \right\},Y} \right\} is a relative topology.

Remark:

Let \left( {X,\tau } \right) be a topological space and Y be the subset of X. Then every open subset 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 a discrete space.