# 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 (relative topology).

Relative Topology or Inherited Topology:

Let $\left( {X,{\tau _X}} \right)$ be a topological space and $Y \subseteq X$ 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 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 member 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 ${\tau _X} = \left\{ {\phi ,\left\{ 2 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,2,3} \right\},X} \right\}$ 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.

Remark:

Let $\left( {X,\tau } \right)$ 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.