“Discreteness” is used in two different equivalent ways

Recently, when thinking about the notion of quantisation in relation to certain differential equations (particularly the Schroedinger equation in quantum mechanics and the ``Road--Wheel equations''), I thought about the meaning of the term ``discrete'' when it comes to usage in common language and in topology.

For example, physicists refer to the solutions for bound states of the time independent Schroedinger equation as having discrete energy levels. In this context, the set of allowed energies is a subset $X\subseteq\mathbb R$ and is said to be discrete in opposition to being ``continuous'', and the intuition is that the discreteness implies we can denumerate the allowable states by assigning a positive integer to each one. However, this raises a nontrivial mathematical problem, namely that it is not immediately obvious how this is equivalent to the topological notion of discreteness.

Problem. If $X\subset\mathbb R$ is discrete (i.e.\@ the inherited subspace topology from $\mathbb R$ is discrete), does this necessarily imply $X$ is countable?

The answer, in alignment with common intuition, is yes.

Proof. We can pick a countable basis $\mathcal B$ for $\mathbb R$ consisting of open intervals with rational endpoints. Since $X$ is discrete, the singleton set $\{x\}$ for any $x\in X$ can be expressed as $X\cap U$ for some $U\in\mathcal{B}$. In this way, we obtain an injection from $X$ into $\mathcal{B}$ by sending each $x$ to the corresponding $U$, so $|X|\leq|\mathcal{B}|=\aleph_0$.

Of course, the proof makes it clear that the topological property of $\mathbb R$ being used here is its second countability (the existence of a countable basis). Results such as this one perhaps have the potential to revive my interest in general topology (or to spark it in the first place), overcoming my initial indifference towards studying countability and separation axioms. We shall see!