Here is Prob. 8, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Which of our four countability axioms does $\mathbb{R}^\omega$ in the uniform topology satisfy?
Here $\mathbb{R}^\omega$ denotes the set of all the (infinite) sequences of real numbers, and the topology in question is that induced by the so-called ''uniform'' metric $\overline{\rho}$ defined on $\mathbb{R}^\omega$ as follows: $$ \overline{\rho} \big( \mathbf{x}, \mathbf{y} \big) := \sup \left\{ \overline{d} \left( x_n, y_n \right) \colon n \in \mathbb{N} \right\}, \tag{A} $$ where $$ \overline{d} \left( x_n, y_n \right) := \min \left\{ \left\lvert x_n - y_n \right\rvert, 1 \right\} $$ for all $$ \mathbf{x} := \left( x_n \right)_{n\in\mathbb{N}}, \mathbf{y} := \left( y_n \right)_{n\in\mathbb{N}} \in \mathbb{R}^\omega $$ [Refer to the Definition just prior to Example 20.4 in Munkres.]
And, our four countability axioms are (i) first-countability, (ii) second-countability, (iii) separability, and (iv) Lindelof-ness.
My Attempt:
(i) First-Countability:
Since our topological space $\mathbb{R}^\omega$ is metrizable, so it is first-countable, by the discussion in Secs. 21 and 30 in Munkres.
Specifically, at any point $\mathbf{p} \in \mathbb{R}^\omega$, the countable collection of all the $\overline{\rho}$-metric open balls of radiuses $1/n$ for all $n = 1, 2, 3, \ldots$ centered at $\mathbf{p}$ is a countable local basis at $\mathbf{p}$.
Am I right?
(ii) Second-Countability:
Let $\mathscr{B}$ be any basis for the uniform topology on $\mathbb{R}^\omega$.
Let $\mathbf{A}$ denote the set of all the sequences of $0$s and $1$s. This set $A$ is uncountable, and for any distinct points $\mathbf{a}, \mathbf{b} \in A$, we have $$ \overline{\rho} (\mathbf{a}, \mathbf{b} ) = 1. \tag{0} $$ So any two open balls of radius $r < 1/2$ centered at $\mathbf{a}$ and $\mathbf{b}$, respectively, are disjoint and open of course, and since there are uncountably many of these balls each having a set from $\mathscr{B}$ contained in it, it follows that the collection $\mathscr{B}$ of sets must also be uncountable.
But $\mathscr{B}$ was an arbitrary basis for the uniform topology on $\mathbb{R}^\omega$. Hence our space is NOT second-countable.
Am I right?
(iii) Separability:
Let $\mathbf{S}$ be any subset of $\mathbb{R}^\omega$ in the uniform topology such that $\mathbf{S}$ is dense in $\mathbb{R}^\omega$. Then for each point $\mathbf{x} \in \mathbb{R}^\omega$ and for each real number $\varepsilon > 0$, there exists a point $\mathbf{s} \in S$ such that $\mathbf{s}$ lies in the open ball of radius $\varepsilon$ centered at $\mathbf{x}$.
In particular, for each point $\mathbf{x}$ in our set $\mathbf{A}$ of all the sequences of $0$s and $1$s and for each radius $r < 1/2$, there exists a point of $\mathbf{S}$ lying within this open ball.
But since there are uncountably many open balls centered at points of $\mathbf{A}$ of radiuses $< 1/2$ --- which are also pairwise disjoint ---, it follows that our set $\mathbf{S}$ must also be uncountable.
But $\mathbf{S}$ was an arbitrary dense subset of $\mathbb{R}^\omega$ in the uniform topology. Hence $\mathbb{R}^\omega$ in the uniform topology is NOT separable.
Am I right?
(iv) Lindelof-ness:
We exhibit an open covering of $\mathbb{R}^\omega$ in the uniform topology that has no countable subcover.
Let $\mathscr{A}$ be the collection of all the open balls centered at points of $\mathbb{R}^\omega$ with radius $< 1/2$.
Since any two distinct points of our set $\mathbf{A}$ of all the sequences of $0$s and $1$s are at a distance of $1$ apart [Refer to (0) above.], no single set in our open covering can contain more than one point of our uncountable set $\mathbf{A}$, thus showing that no countable subcollection of $\mathscr{A}$ can cover $\mathbf{A}$ and hence all of $\mathbb{R}^\omega$.
Hence $\mathbb{R}^\omega$ in the uniform topology is NOT Lindelof.
Am I right?
Is each one of my solutions above correct and clear enough in each and every detail? Or, have I made a mistake somewhere without my noticing it?
As far as I can see it, my conclusions, as well as the arguments I've given in support of these conclusions, are over all correct. However, it sometimes happens that I commit mistakes without realising it! Hence the rather too broad the scope of my question in the preceding paragraph.
