The original problem asks the following: \begin{equation} \text{Prove that } \mathbb{R}^\mathbb{R} \text{has a subspace } X \text{ that is } T_4 \text{ but not Lindelöf}. \end{equation}
Additionally, I was given the following hint: \begin{equation} \text{ Start by proving } \{f_r \text{ | } r \in \mathbb{R}\} \text{ where } f_r(x) = 1 \text{ if } x=r \text{ else } 0 \text{ is discrete.} \end{equation}
Thus far, I know my proof has to take the following steps:
(1) Prove $\{f_r \text{ | } r \in \mathbb{R}\}$ is discrete.
(2) Since said subspace $X$ is discrete, it satisfies all the separation axioms.
(3) Since it is uncountable discrete, it is not Lindelöf.
The main snag I hit is in step 1. If any other steps in what I perceive to be the proof process are wrong, feel free to correct me/elaborate!
Thanks in advance.
Hint: Let $\pi_r:\Bbb R^{\Bbb R}\to\Bbb R$ be the '$r$th projection', i.e. the evaluation at $r$, and consider the open set $\pi_r^{-1}(\frac12,\frac32)$.