$[0,\omega_1]$ is not first countable.
97053
To prove $\omega_1 \cup \{\omega_1\} $ is not first countable but $\omega_1$ is
62793
$\mathbb{R}$ with the lower limit topology is not second-countable
27937
Cartesian Product of Two Countable Sets is Countable
128118
Is the cofinite topology on a countable set first countable?
93389
Uncountable sets of Hausdorff dimension zero
139247
Real Analysis - Prove a set is countable
156978
First uncountable ordinal
11959
What's wrong with this argument that $[0,1]$ is countable?
154474
Image of countable and uncountable sets
149822
If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$?
93606
Showing that $C[0,1]$ is not complete
136538
How can I show that a bijection between $[0,1)$ and $(0,1)$ cannot be continuous?
127362
How do I prove that there doesn't exist a set whose power set is countable?
134575
Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.
128437
How to prove that a finite set is countable
40937
Is the set of partial order relations over $ \mathbb{N}$ countable?
42095
Existence in ZF of a set with countable power set