Let $\mathbb{R}^\mathbb{N}_\text{prod}$ be countable product of $\mathbb{R}$ with the product topology, $\mathbb{R}^\mathbb{N}_\text{prod}$ be countable product of $\mathbb{R}$ with the uniform topology. I wish to show that: Show that $\mathbb{R}^\mathbb{N}_\text{prod}$ has the countable cha...

The tag metric-topology has been created recently. Currently it contains only a few questions. As far as I can say from the questions currently tagged with this tag, they are about the topology derived from a metric. In my opinion, the already existing tag metric-spaces can be used for such ques...

Theorem: (Baire) $(X,d)$ is a complete metric space then the intersection of countably many dense, open subsets in the metric topology $\mathcal{T}$ generated by $d$ is dense In other words, $$D = \bigcap_{n \in \mathbb{N}} D_n$$ where $D_n$ is a dense, open subset of $X$, i...

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from closed set $C$ using disjoint open sets $U,V \in \mathcal{T}$. Show that any metrizable space is r...

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The metric on the product topology on $\mathbb{R}^\mathbb{N}$ is assumed to be: $$d_p(x,y) = \sup_{n...

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that the box topology is not metrizable, so we cannot compare metric balls... All is not lost, because ...

Let metric topology be the topology generated by metric balls of a metrizable space $X$ Is there a subbase $S$ that generates the metric topology? I am asking because in most textbooks, it seems that people only discuss the base generated by the metric balls, and no attention is paid to subbase...

This is a follow up to one of my earlier questions I am reading some stuff online and saw a proof as follows Per a comment in Part 1 in linked, We know that $d(C_1,C_2)$ could easily be zero for disjoint closed sets (Intuition: Think Gabriel's Horn). Does the union of all the $B_{\frac{d}{...

In these notes I came across a curious notation Let $\overline d(x,y) = \min\{d(x,y), 1\}$ is a well known standard bounded metric on a metric space $(X,d)$ However, in the notes in linked, it was rewritten as: Given any metric space $(X, d)$ $\overline d(x,y) = d(x,y) \wedge1$, where $...

Let $\mathbb{R}^\mathbb{N}_\text{prod}$ be countable product of $\mathbb{R}$ with the product topology, $\mathbb{R}^\mathbb{N}_\text{prod}$ be countable product of $\mathbb{R}$ with the uniform topology. I wish to show that: Show that $\mathbb{R}^\mathbb{N}_\text{prod}$ has the countable cha...