Topological Subspaces Examples 1

Topological Subspaces Examples 1

Recall from the Topological Subspaces page that if $(X, \tau)$ is a topological space and $A \subseteq X$ then the subspace topology on $A$ is defined to be:

(1)
\begin{align} \quad \tau_A = \{ A \cap U : U \in \tau \} \end{align}

We verified that $\tau_A$ is indeed a topology for any subset $A$ of $X$.

We will now look at some examples of subspace topologies.

Example 1

Consider the topological space $(\mathbb{R}^2, \tau)$ where $\tau$ is the usual topology of open disks in $\mathbb{R}^2$. Determine what the subspace topology is for the subset $A = \{ (x, 0) \in \mathbb{R}^2 : x \in \mathbb{R} \} \subseteq \mathbb{R}^2$.

Note that the set $A = \{ (x, 0) \in \mathbb{R}^2 : x \in \mathbb{R} \}$ is simply the real line $\mathbb{R}$. Geometrically we can see that the subspace topology $\tau_A$ will simply be the usual topology on $\mathbb{R}$.

To see this, consider any open set in $\mathbb{R}$ with respect to the usual topology of open intervals in $\mathbb{R}$. Then any open interval $(a, b)$ can be constructed by taking an open disk in $\mathbb{R}^2$ that intersects the line $y = 0$ at the points $(a, 0)$, $(b, 0)$. Since every open set in $\mathbb{R}$ is a union of these open intervals, we see that the subspace topology on $\mathbb{R}$ is simply the usual topology on $\mathbb{R}$.

Example 2

Consider the topological space $(\mathbb{R}, \tau)$ where $\tau$ is the usual topology of open intervals in $\mathbb{R}$. Verify that the subspace topology on $\mathbb{Z} \subseteq \mathbb{R}$ is the discrete topology on $\mathbb{Z}$.

Let $x \in \mathbb{Z}$. Then the open interval $\left ( x - \frac{1}{2}, x + \frac{1}{2} \right) \cap \mathbb{Z} = \{ x \}$. Hence every singleton set $\{ x \}$ is contained in the subspace topology on $\mathbb{Z}$. But this implies that that $\tau_{\mathbb{Z}}$ is the discrete topology on $\mathbb{Z}$.