Hereditary separability/Lindelöfness and bounds for size of spaces. I'm reading a proof that seems to assume the following statements:

$X$ $T_2$ and hereditarily Lindelöf $\implies$ $|X|\leq 2^{\aleph_0}$
$X$ $T_3$ and hereditarily separable $\implies$ $w(X)\leq 2^{\aleph_0}$
$w(X)\leq 2^{\aleph_0}$ and X hereditarily Lindelöf $\implies$ $|\tau|\leq 2^{\aleph_0}$ where $\tau$ is the topology of $X$.

Are these folklore? Or are they simple? Thank you for any solutions/references.
 A: $X$ is $T_3$ and separable then $w(X) \le 2^{\aleph_0}$, because the regular open sets $RO(X)$ of $X$, i.e. all open sets $O$ such that $O = \operatorname{int}(\overline{O})$, for $A \subseteq X$, are a base for $X$. (This follows from $T_3$: $x \in O$ and $O$ open, then there is some $V$ open with $x \in V \subseteq \overline{V} \subseteq O$, and then $\operatorname{int}(\overline{V})$ is regular open and sits between $x$ and $O$). 
Moreover, if $D$ is countable and dense, and $O$ is regular open, then $O = \operatorname{int}(\overline{D \cap O})$, so every regular open set is uniquely determined by a subset of $D$ (namely $D \cap O$), or equivalently the mapping $O \to O \cap D$ from $RO(X) \to \mathcal{P}(D)$ is 1-1 and the latter set has size $2^{\aleph_0}$, so $w(X) \le |RO(X)|$ and we are done.
If $X$ is hereditarily Lindelöf, and $\mathcal{B} = \{B_i: i \in I\}$ is a base of size $\kappa \le 2^{\aleph_0}$, then for every open set $O$, we can write $O = \cup \{B_i: i \in I_O \}$ for some $I_O \subseteq I$. But as $X$ is hereditarily Lindelöf (hence $O$ is Lindelöf), we have some countable subset $N_O \subseteq I_O$ such that $O = \cup\{B_i: i \in N_O\}$. So we have a 1-1 map from $\tau$ to all countable subsets of $I$, namely $O \to N_O$.
Hence $$|\tau| \le \kappa^{\aleph_0} \le (2^{\aleph_0})^{\aleph_0} = 2^{\aleph_0 \cdot \aleph_0} = 2^{\aleph_0}$$
as required.
The fact that hereditarily Lindelöf Hausdorff spaces have size at most $2^{\aleph_0}$ is due to de Groot (Discrete spaces of Hausdorff spaces. Bull. Acad. Polon. Sci. 13, 537-544) and is Corollary 4.10 in Hodel's chapter (Cardinal functions I) in the Handbook of Set-Theoretic Topology, where it is derived from more general bounds on $|X|$. It's a bit too long to repeat here. It's also theorem 2.5 in Juhasz' book Cardinal Functions in Topology (where it has a rather direct proof). 
So in summary: nr 1 is a classic published result, needing some set theory machinery. 2 only needs separable and is "folklore" (though it is mentioned on Hodel's article and Juhasz' book as well), and 3. is a standard counting argument.
A: They are all special cases of the following cardinal inequalities:


*

*$|X| ≤ 2^{h(X)}$ for every $T_2$ space $X$ where $h(X)$ is the hereditarily Lindelöf number.

*$w(X) ≤ 2^{d(X)}$ for every $T_3$ space $X$.

*$o(X) ≤ w(X)^{h(X)}$ for every space $X$ where $o(X)$ is the cardinalily of the topology.


I would say they are folklore (at least the last two), and they can all be found in the Juhasz's book on cardinal functions in topology.
Some of the proofs are quite simple. And now I can see Henno Brandsma has already written them down in his answer.
