I am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of basis elements of the form $\{[a,b)\mid a<b, a,b\in \mathbb{R}\}$.
These are some of the properties for $\mathbb{R}_l$ mentioned in the book.
1) It is Hausdorff
2) It is first countable but not second countable
3) It is normal
Here is my understanding of the proofs for these properties.
1) $\mathbb{R}_l$ is Hausdorff as if I consider two points $x$ and $y$ with $x<y$, I can find disjoint open sets $[x,y)$ and $[y,y+\delta)$ containing $x$ and $y$, respectively.
2) It is first countable as every point $x$ in $\mathbb{R}_l$ has a countable basis, but I don't understand why it is not second countable, as it should have countable basis for this topology. Is it because if I consider an open set for the standard topology on $R$, say $(a,b)$, this set is an union of uncountable open sets in $\mathbb{R}_l$
$(a,b)=\cup_{a<x<b}[x,b)$
3) I understand the proof given in the book for $\mathbb{R}_l$ to be normal.
Please tell me if my understanding of these concepts is correct. I would appreciate if someone can provide more deeper insight about these concepts.
Assume $X=(\Bbb R,\tau_l)$ were second-countable. One can prove that in a second countable space, every base $\cal B$ contains a countable base, so we can assume that the countable base $\cal C$ has only sets of the form $[a,b)$ for $a<b$, and we can enumerate them as $[a_n,b_n)$. Now take a point $x$ distinct from any $a_n$. Then there is no neighborhood of $x$ in $\cal C$ contained in the neighborhood $[x,\infty)$. That means $\Bbb R$ is not second countable.
For the Hausdorffness, you can give a direct argument, as you did here, or you can use the fact that the lower limit topology is finer that the usual topology on $\Bbb R$ which is Hausdorff. For normality, however, this doesn't work as a finer topology can cease to be normal.