Euclidean Space is Second-Countable

Theorem

Let $\R^n$ be an $n$-dimensional real vector space.

Let $\struct {\R^n, \tau_d}$ $n$-dimensional real Euclidean space with the usual topology.

Then, $\struct {\R^n, \tau_d}$ is second-countable.

Proof

Let $\struct {\R, \tau}$ be the real line with usual topology.

Let $T_n = \struct {\R^n, \tau_n}$ be the topological space such that $\tau_n$ is the product topology on $\R^n$.

From Real Number Line is Second-Countable, we have that $\struct {\R, \tau}$ is second-countable.

From Countable Product of Second-Countable Spaces is Second-Countable, we have that $\struct {\R^n, \tau_n}$ is second-countable.

From Euclidean Topology is Product Topology, we have that $\tau_d = \tau_n$.

Therefore, $\struct {\R^n, \tau_d}$ is second-countable.

$\blacksquare$