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.



Let $$X = \left\{ {1,2,3,4} \right\}$$ with topology $${\tau _X} = \left\{ {\phi ,\left\{ 2 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,2,3} \right\},X} \right\}$$ and$$Y = \left\{ {1,3,4} \right\} \subseteq X$$, 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.


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.