This might be an easy question, maybe the example I'm looking for is common knowledge. As always, recall that a topological space $X$ is *scattered* if and only if every non-empty subset $Y$ of $X$ contains at least one point which is isolated in $Y$. It is known that any first countable, $T\_3$, Lindelöf and scattered space is countable. Is there an example of an uncountable, first countable, Hausdorff, Lindelöf and scattered space?



Example of an uncountable scattered space with some properties
There exists an example for this question under Continuum Hypothesis, more precisely, under the assumption $\mathfrak b=\omega\_1$. In this case by Theorem 10.2 in the van Douwen's survey paper ``[The integers and Topology](https://www.sciencedirect.com/science/article/pii/B9780444865809500069)'', there exists an uncountable set $C\subseteq\mathbb R\setminus\mathbb Q$, which is *concentrated* at rationals in the sense that for every open set $U\subseteq\mathbb R$ with $\mathbb Q\subseteq U$ we the complement $C\setminus U$ is countable.


Consider the uncountable space $X=\mathbb Q\cup C$ endowed with the topology $\tau$ consisting of the sets $W\subseteq X$ such that for every $q\in W\cap\mathbb Q$ there exists $\varepsilon>0$ such that $\{x\in C:|x-q|<\varepsilon\}\subseteq W$.


It is easy to see that the space $(X,\tau)$ is (functionally) Hausdorff, first-countable, Lindelof and scattered. More precisely, the subspace $C=X\setminus\mathbb Q$ is open and discrete and $\mathbb Q$ is closed and discrete in $X$. So, $X$ is scattered of finite scattered height.


To see that $X$ is Lindelof, observe that for every open cover $\mathcal U$ there exists a countable subfamily $\mathcal V\subseteq\mathcal U$ such that $\mathbb Q\subseteq \bigcup\mathcal V$. Since $C$ is concentrated at $\mathbb Q$, the difference $C\setminus\bigcup\mathcal V$ is countable and hence we can choose a countable subfamily $\mathcal V'\subseteq \mathcal U$ such that $C\setminus \bigcup\mathcal V\subseteq\bigcup\mathcal V'$. Then $\mathcal V\cup\mathcal V'$ is a required countable subcover of $\mathcal U$, witnessing that $X$ is Lindelof.




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The above example cannot be constructed in ZFC because of the following


**Theorem.** Under $\mathfrak b>\omega\_1$, every scattered Lindelof $T\_1$-space of countable pseudocharacter and finite scattered height is countable.


Prior writing down the proof of this theorem, let us recall the definition of the scattered height.


Given a subset $A$ of a topological space $X$, let $A^{(1)}$ be the set of nonisolated points of $A$. Observe that for a closed subset $A$ of $X$, the set $A^{(1)}$ is closed in $A$. Let $X^{(0)}=X$ and for every nonzero ordinal $\alpha$ let
$$X^{(\alpha)}=\bigcap\_{\beta<\alpha}\big(X^{(\beta)}\big)^{(1)}.$$
By transfinite induction it can be shown that $(X^{(\alpha)})\_\alpha$ is a decreasing sequence of closed subsets of $X$, so it should stabilize at some ordinal $\alpha$. The smallest ordinal $\alpha$ such that $X^{(\alpha+1)}=X^{(\alpha)}$ is called the *scattered height* of $X$ and is denoted by $\hbar(X)$. A topological space $X$ is scattered if and only if $X^{(\hbar(X))}=\emptyset$.


*Proof of Theorem.* Assume that $\mathfrak b>\omega\_1$. By Mathematical Induction we shall prove that *for every natural number $n$, every Lindelof scattered space $X$ of countable pseudocharacter and scattered height $\hbar(X)\le n$ is countable.*


For $n=0$, each scattered space $X$ with $\hbar(X)\le 0$ is empty and hence countable. Assume that for some positive integer number $n$, every Lindelof scattered space $X$ of countable pseudocharacter and scattered height $\hbar(X)<n$ is countable. Take a Lindelof scattered space $X$ of countable pseudocharacter and scattered height $\hbar(X)=n$. Then $X^{(n)}=\emptyset$ and $X^{(n-1)}$ is a nonempty discrete subspace of $X$. Being a closed subspace of the Lindelof space $X$, the discrete space $X^{(n-1)}$ is Lindelof and countable. Write $X^{(n-1)}$ as $\{x\_k\}\_{k\in\omega}$. Since $X$ has countable pseudocharacter, for every $k\in\omega$ there exists a decreasing sequence $(U\_{k,m})\_{m\in\omega}$ of open sets in $X$ such that $\bigcup\_{m\in\omega}U\_{k,m}=\{x\_k\}$. Assuming that the space $X$ is uncountable, choose a transfinite sequence $\{y\_\alpha\}\_{\alpha\in\omega\_1}$ consisting of pairwise disjoint elements of $X\setminus X^{(n-1)}$. For every $\alpha\in\omega\_1$, let $f\_\alpha:\omega\to\omega$ be the function assigning to each $k\in\omega$ the smallest number $f\_\alpha(k)$ such that $y\_\alpha\notin U\_{k,f\_\alpha(k)}$. The number $f\_\alpha(k)$ exists since $y\_\alpha\notin\{x\_k\}=\bigcap\_{m\in\omega}U\_{k,m}$. Since $\mathfrak b>\omega\_1$, there exists a countable subfamily $\mathcal F\subseteq\omega^\omega$ such that for every $\alpha\in\omega\_1$ there exists a function $f\in\mathcal F$ such that $f\_\alpha\le f$. For every $f\in\mathcal F$, consider the open neighborhood $U\_f=\bigcup\_{k\in\omega}U\_{k,f(k)}$ of the set $\{x\_k\}\_{k\in\omega}=X^{(n-1)}$. The choice of $\mathcal F$ guarantees that $\{y\_\alpha\}\_{\alpha\in\omega\_1}\cap\bigcap\_{f\in\mathcal F}U\_f=\emptyset$. By the Pigeonhole Principle, for some $f\in\mathcal F$, the set $\Omega=\{\alpha\in\omega\_1:y\_\alpha\notin U\_f\}$ is uncountable. On the other hand, $Y=X\setminus U\_f$ is a closed (and hence Lindelof) subspace of $X$. It follows from $Y\cap X^{(n-1)}=\emptyset$ that $\hbar(Y)<n$ and hence $Y$ is countable, by the inductive hypothesis. But this contradicts the uncountability of the set $\{y\_\alpha\}\_{\alpha\in\Omega}\subseteq Y$.




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Can the theorem be generalized to scattered spaces of arbitrary scattered height?


**Problem.** *Assume that $\mathfrak b>\omega\_1$. Is every scattered Lindelof $T\_1$-space of countable pseudocharacter countable?*




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**Remark.** Thanks to the intervention of Will Brian, I realized that the equality $\mathfrak d=\omega\_1$ in the initial redaction of my answer can be replaced by the weaker equality $\mathfrak b=\omega\_1$. So, now we have the following characterization.


**Theorem.** *The following conditions are equivalent:*


1. $\mathfrak b>\omega\_1$.
2. *Every Lindelof scattered $T\_1$-space of countable pseudocharacter finite scattered height is countable.*
3. *Every first-countable Lindelof scattered Hausdorff space of scattered height 2 is countable.*


The answer of Will Brian shows that this theorem does not extend to scattered Lindelof spaces of infinite scattered height.


