Let $\tau$ be lower limit topology on $\mathbb R.$ Then $(X,\tau)$ is not second countable.
62793
$\mathbb{R}$ with the lower limit topology is not second-countable
95635
$\ell^{\infty}(\mathbb N)$ is not a separable space
16649
Topological space with countable open cover $\{U_\alpha\} $ with each $U_\alpha$ second-countable, is second countable
150796
Subspace topology and order topology (2)
7607
How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
27937
Cartesian Product of Two Countable Sets is Countable
130361
Basis for a topology with a countable number of sets
84460
If a set is $F_\sigma$ and is of first category then it is a countable union of closed nowhere dense sets
70413
A vector space over $R$ is not a countable union of proper subspaces
13620
$\omega_2$ is a not countable union of countable sets without AC
106298
Prove that some topology is not metrizable
71173
Show that $\mathbb{A}^n$ on the Zariski Topology is not Hausdorff, but it is $T_1$
128766
Show that the set of functions $\mathbb{N}\to\{0,1\}$ is not countable
101986
Prob. 2, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: The dictionary order topology on $\mathbb{R} \times \mathbb{R}$ is metrizable.
120135
Show every nonempty compact Hausdorff space is not the countable union of nowhere dense sets
80414
Countable Product of Sequentially Compact spaces is Sequentially Compact
47251
Non-compactness of $\mathbb{R}$ with the cocountable topology