# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### Alternative proof: show that any metrizable space $X$ is normal - Part 1

There is a proof online that shows that all metric spaces are normal. The proof is as follows However, it has the additional baggage of needing to show that $d(x,A)$ is continuous and $U,V$ are ...
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### Are symmetric and $\Delta$-metric common terminologies?

In these notes on metric spaces, the author also defined something known as "symmetric", and $\Delta$-metric. I have never seen these terminologies before. Are these terms standard usage? Can ...
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### $X$ be real i.p.s. dim.>1 , if two closed balls,none of which is a subset of the other,intersect then do the boundaries of the balls intersect too?

Let $X$ be a real inner product space of dimension more than $1$ , let $B[x;r] , B[y;s]$ be two closed balls having non-empty intersection where none of the balls is a subset of the other , then is ...
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### $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ via $x \mapsto\frac{x}{\|x\|}$ is continuous

I'm having trouble understanding why a map $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ (unit $n$-sphere, $n\ge 1$) via $x \mapsto\frac{x}{||x\|}$ is continuous. Since Unit n-Sphere under Euclidean ...
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### Show $A\cap B \neq \varnothing \Rightarrow \operatorname{dist}(A,B) = 0$, and $\operatorname{dist}(A, B) = 0 \not\Rightarrow A\cap B \neq \varnothing$

I have a question Let $d$ be a metric on $X$, and define the set to set distance $$\operatorname{dist}(A,B) = \inf\{d(x,y): x\in A, y \in B\}$$ where $A,B \subseteq X$ are nonempty sets ...
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### Diameter of set in metric space

I do agree with the statement that $$d(A) = \sup{\{d(x, y):x, y \in A\}}$$ But why can't we use maximum because according to me its max will also give diameter. I know it should not be correct, so ...
According to the nLab, Lawvere metric spaces occur rather naturally (that is as certain enriched categories). A Lawvere metric space is a set $X$ equipped with a function $d : X\times X \to [0,\infty]$...