The definition I am working with is that a topological space is Luzin if it is an uncountable $T_3$ space such that every nowhere dense subset is at most countable.
I've been reading the proof of Theorem 6.8 from S. Todorcevic's Partition Problems in Topology: If it is true that every regular hereditarily Lindelof space is hereditarily separable, then there are no Luzin spaces. The proof proceeds by contradiction, assuming there is a Luzin space $X$ and building from it a hereditarily Lindelof space that is not hereditarily separable. Earlier in the book, it is proved that the same hypotheses imply the Suslin Hypothesis, i.e. there are no Suslin trees ($\omega_1$-trees with no uncountable chains and no uncountable antichains).
At some point in the proof, the author claims that since there are no Suslin trees, in particular, the set of nonempty open sets of $X$ ordered by set-theoretic inclusion is not a Suslin tree. Therefore it is possible to find a continuous function from an uncountable subset $X_0\subseteq X$ into the reals, $f: X_0\rightarrow \mathbb{R}$, such that $f$ is one-to-one. Finding such a function is the very assertion that I am struggling to see.
Since there is no reference to previous results or bibliography about this assertion, I believed it is folklore or something very easy to see. However, I have been struggling to find a proof for this.
What I have done: First, I found this theorem in K. Kunnen's paper "Luzin spaces" http://www.topology.auburn.edu/tp/reprints/v01/tp01021.pdf : Every Luzin ($T_3$) space is zero-dimensional (in the paper, a Luzin space is a $T_2$ topological space such that every nowhere dense subset is countable). Because of this, instead of considering the whole set of nonempty open subsets of $X$, I rather work with the clopen basis, ordered by inclusion.
Secondly, every Luzin space is hereditarily Lindelof, and by our hypothesis, also hereditarily separable. So the clopen basis ordered by inclusion is $ccc$. Since it is not a Suslin tree, then either it's height as a tree is countable or there is an uncountable chain. If there is an uncountable chain $\{U_\alpha:\alpha<\omega_1\}$, then $\bigcup\{X\setminus U_\alpha:\alpha<\omega_1\}$ is not hereditarily Lindelof, for the family $\{X\setminus U_\alpha:\alpha<\omega_1\}$ is an open cover with no countable subcovers. Thus, the tree is countable.
From this, I tried to build a continuous one-to-one function from the tree $2^{<\omega}$ into the reals, enumerate the countable dense subset of $X$, $D=\{d_n:n\in \omega\}$, and then associate to every $x\in X$ an infinite 0,1 sequence, such that the sequence is the characteristic function of a subset of $D$ converging to $x$. The only problem here is that I don't know whether $X$ is first countable, or at least Frechet-Urysohn. I believed it was true, for if there was a point $x$ with an uncountable local base, then one should be able to build an uncountable chain of basic clopens of $x$. However, I am not sure if I am thinking of this in the wrong way.
I then pursued another strategy: Using the clopen basis, it is easy to build a Cantor scheme $(A_s,\subseteq)$, $s\in 2^{<\omega}$, without the vanishing diameter condition, since we don't know about metrizability in $X$. Nevertheless, if it is true that the set of chains of this scheme whose intersection is non-empty, then it is possible to define a continuous one-to-one function like the one we want. In other words, if the set $P=\{f\in 2^\omega:\bigcap_{n\in\omega}A_{f|_n}\neq\emptyset\}$ is uncountable, we are done. But I don't know if this is true.
Looking up for answers, I found the concepts of "Suslin representation" and "Suslin operation", which seem to address similar problems. However, they seem to involve too much infinitary combinatorics for Todorcevic to not refer to a particular paper, book, or result that implies the assertion.
Is that assertion trivial and I am completely lost? Or is there a reference where to look up for this result?