Subspace topology

The subspace topology on a subset $Y\subseteq X$ of a topological space $(X,\mathcal{T})$ is the topology

With this topology, $(Y,\mathcal{T}_Y)$ becomes a topological space, called a subspace of $X$.

A subset $V\subseteq Y$ is open in the subspace exactly when it is the intersection of $Y$ with an open set of $X$. The inclusion map $i:Y\to X$ is automatically continuous for the subspace topology.

Examples:

• $[0,1]\subseteq \mathbb{R}$ with the subspace topology has open sets of the form $U\cap[0,1]$, where $U$ is open in $\mathbb{R}$.
• If $Y=\{x_0\}$ is a singleton subset of any space $X$, then $Y$ has the indiscrete (and also discrete) topology $\{\varnothing,Y\}$.
• Any subset of a discrete space is discrete in the subspace topology.