9:56 AM
Somebody created tag.
1

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...

Since there already exists , I am going to replace it. I have also suggested a synonym.
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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...

There are 9 questions tagged at the moment:
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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...

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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...

1

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive semidefiniteness and symmetry are trivial We want to show that $$d_2(f,g) \leq d_2(f,h) + d_2(... 1 I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on \mathbb{R} denoted \mathbb{R}^\mathbb{N} We know both spaces are metrizable: The metric on the product topology on \mathbb{R}^\mathbb{N} is assumed to be:$$d_p(x,y) = \sup_{n...

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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 ...

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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...

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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 $B_{\frac{d}{... 0 In these notes I came across a curious notation Let$\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$, where$...

1

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...

I have also pinged the (probable) tag-creator:
@BeachedWhale It seems very likely that you are the creator of (metric-topology) tag. I wanted to let you know that I have made a post on meta to discuss this new tag. — Martin Sleziak 57 secs ago

1 hour later…
11:20 AM
@MartinSleziak Hi! Yes I like to create new tags at whim, such as Arzela Ascoli. I created metric topology because this is now used a lot in formal teaching as people confuse metric space with topological space and vice versa, mainly because real analysis and topology are taught separately. For example, most people are not introduced to formal definition of topology in real analysis, therefore all intuition about convergence of sequences etc are wrt metric topology, but does not hold in general topology. Perhaps metrizable spaces is more appropriate. But do whatever as you wish to the tag! — Beached Whale 32 mins ago

1 hour later…
12:24 PM
Thanks for the response. My impression is that the tag (metric-spaces) is typically used for questions related to metric space, metrizable spaces, topology derived from a metric etc. In any case, if we want to continue this discussion, it would be probably more suitable on meta or in chat. — Martin Sleziak 32 secs ago

1 hour later…
1:29 PM
Is the tag going to stay? Was is discussed somewhere?
Probably many it would be a good tag for many of the questions currently tagged matrices+exponential-function.

6 hours later…
7:24 PM
is now at 190 questions.