Locally Compact Haudroff First countable space which are not Lindelolf.
Locally Compact Hausdroff first countable and seperable which is not Lindelof.
Locally Compact Hausdroff first countable and Lindelof space which is not seperable.
Locally compact Haudroff space which is not normal
(if exists)
An uncountable discrete space.
(1. doesn't work any more) Mrówka $\Psi(\mathcal{A})$-space, for any almost disjoint family $\mathcal{A}$. See my post here for a definition.
The lexicographically ordered square $[0,1] \times [0,1]$ in the order topology is compact Hausdorff (so Lindelöf and locally compact follow ) but not separable. A similar example is $\omega_1 + 1$ in the order topology. Both are (hereditarily) normal though. Other large compact spaces like the product $[0,1]^I$ for $|I| > \mathfrak{c}$, and $\beta \omega$ are also examples.
A Mrówka space $\Psi(\mathcal{A})$ where $\mathcal{A}$ is a maximal almost disjoint family of $\mathcal{P}(\omega)$, because then this spacae is pseudocompact but not countably compact, so cannot be normal. Other example (more advanced) is $\beta \omega \setminus \{p\}$ where $p \notin \omega$. The classic example (besides the deleted Tychonoff plank) is $\alpha(N) \times \alpha(D) \setminus \{(\infty_1, \infty_2)\}$, where $\alpha(N)$ is the one-point compactification $N \cup \{\infty_1\}$ of a countable discrete space $N$ and $\alpha(D)$ the one-point compactification $D \cup \{\infty_2\}$ of an uncountable discrete space $D$.
The now classic resource (which I did not yet use so far) for questions like this is $\pi$-base, based on the even more classic book Counterexamples in Topology by Steen and Seeabach, and which you can query, e.g. for question 1 above to indeed see the example that first came to my mind too. The rational sequence topology there is also an extra example for 4 and 2 as well (Mrówka space is not in the database yet).
