I want to find an example (or the insight that there is no such set) of closed nowhere set with no isolated points from the left. I tried to write down some constructions of which I wasn't able to show closedness, for example the naive construction $A := \cup_k A_k$ with $A_0 := \{1\}$ and
$$A_k := \bigcup_{a\in A_{k-1}} \{a\}\cup \{a - \frac 1 n : n\in \Bbb N\}$$
Without the restriction of isolation from the left a good example would be of course the cantor set $C$ which is known to be closed and to not contain isolated points. But for example $\frac 2 3 \in C$ but $( 2/3 - 1/6 , 2/3 ) \cap C = \emptyset$. Thus $2/3$ is isolated from the left.
It seems not so easy how to adapt the cantor set construction to this problem. But I am not really experienced with this type of problems. Moreover, this appears to me as already known somewhere. A sufficient reference would be also an answer to my question.
The empty set works. It is nowhere dense and has no points at all; in particular it has no points isolated from the left.
Now suppose $C \subseteq \mathbb R$ is nonempty closed and nowhere dense in $\mathbb R$. There is a point $a \in C$. Then $C_L := (-\infty , a)\setminus C$ is open. $C$ is nowhere dense, so $C_L \ne \varnothing$.
An open set is the disjoint union of a countable family of open intervals. So:
There exist $-\infty \le b < c \le a$ with $(b,c)$ a maximal interval in $C_L$, and therefore $(b,c)$ is disjoint from $C$. Then (by maximality) $c \in C$ so it is isolated from the left.