Discrete space $(X, \tau)$ is homeomorphic to a subspace of $\mathbb{R}$ iff $X$ is countable

Question

Let $\mathbb{R}$ have the Euclidean topology (i.e., with Euclidean distance), and let $(X, \tau)$ be a discrete topological space.

Prove that $(X, \tau)$ is homeomorphic to a subspace of $\mathbb{R}$ iff $X$ is countable.

Let $X$ be countable. If $|X|$$=n$, then the subspace $A=\{1,2,3,...,n\}$ with the induced Euclidean topology is homeomorphic to $(X, \tau)$. If X is countably infinite, then the subspace $\mathbb{N} $ is homeomophic to $(X, \tau)$.

Now how we can prove the other direction, i.e., if $(X, \tau)$ is homeomorphic to a subspace of $\mathbb{R}$, then $X$ is countable?

Answer 1

HINT: Prove the contrapositive: show that if $A\subseteq\Bbb R$ is uncountable, then the relative topology on $A$ is not discrete. One way to do this is to let $\mathscr{B}$ be a countable base for the topology of $\Bbb R$, and let

$$\mathscr{B}_0=\{B\in\mathscr{B}:B\cap A\text{ is countable}\}\;.$$

• Show that $A\cap\bigcup\mathscr{B}_0$ is countable.

Let $A_0=A\setminus\bigcup\mathscr{B}_0$.

• Show that $A\ne\varnothing$, and that each $x\in A_0$ is a limit point of $A$. Conclude that $A$ is not discrete in its relative topology.