Countability Axioms - finer/coarser topology

Question

Beginner in point-set topology here - since I can't verification of the below statements, I wonder if the following are correct.

If $X$ is first countable in the finer topology, $X$ is not necessarily first countable in the coarser topology. ($\mathbb{R}^{\omega}$ in the uniform topology is first countable, but it's not first countable in the box topology. Similarly, first countable in coarser topology doesn't imply first countable in finer topology, as in the case of $\mathbb{R}^{\omega}$ under the product and box topology)

However, if $X$ is second countable in the finer topology, $X$ is first countable in the coarser topology, because the basis in the finer topology serves as a basis for the coarser topology.

Answer 1

The first example: $\mathbb{R}^\omega$ uniform is $C_I$, $\mathbb{R}^{\omega}$ in the box topology not $C_I$ only shows we cannot go from a coarser to a finer topology w.r.t. $C_I$, not as you claim, the other way around.

The last paragraph is completely false: take a countable discrete space which is $C_I$ and $C_{II}$, but there are countable spaces with weight and character $\mathfrak{c}$, so we cannot say anything sensible about a coarser topology on the same set.

In other words, you cannot make any claim w.r.t. $C_I$ an $C_{II}$ either from coarser to finer or finer to coarser.