I am trying to solve
What are the path components of $\mathbb{R}_l$?
We know that $\mathbb{R}_l$ is lower limit topology.
Definition of path component here
I am thinking that the path components of $\mathbb{R}_l$ are singleton sets.
MY ATTEMPT:
Suppose it is not, let $A$ be a path connected subspace of $\mathbb{R}_l$ with more than one point. Let $a, b \in A$ such that $a<b$. Then the sets $A \cap (-\infty, b)$ and $A \cap [b, +\infty)$ are open in $A$, and clearly $a \in A \cap (-\infty, b)$ and $b \in A \cap [b, +\infty)$, hence nonempty, and also $A = (A \cap (-\infty, b)) \sqcup (A \cap [b, +\infty))$
But I have no idea how to proceed further. Please help me.
You are right: the only connected subsets of $\Bbb R_l$ are the singletons; in other words, $\Bbb R_l$ is totally disconnected.
Take $a,b\in\Bbb R$, with $a<b$, and suppose that there is a connected subset $C$ of $\Bbb R_l$ such that $a,b\in C$. Let $c\in(a,b)$. Then both $[c,\infty)$ and $[a,c)$ are clopen sets. In particular, $[c,\infty)\cap C$ and $[a,c)\cap C$ are non-empty disjoint subsets of $C$ whose union is $C$. This is impossible, since $C$ is connected.