# 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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### Proving this function is continuous and bounded

Let $(X,d)$ be a metric space. Fix $a \in X$. For every $x \in X$ we consider the function $$f_x : X \rightarrow \mathbb{R}: y \mapsto f_x(y) = d(x,y) - d(a,y).$$ Prove that $f_x$ is continuous ...
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### In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
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### I am searching for a metric space in which every Cauchy sequence is stationary.

I am searching for an example of a metric space in which every Cauchy sequence is stationary, means that there exists some $N$ such that for all $n>N$: $a_n = a_N$. Is there a simple example for ...
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### surface to surface map, $f$ is closed but neither open nor continuous

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
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### Proving triangle inequality using complete-linkage between clusters and arbitrary dissimilarity measure

Assuming a dissimilarity measure d satisfies the usual properties, I need to prove that complete linkage ( i.e. d(A,B)=maxx∈A,y∈B{d(x,y)} ) either satisfies or does not satisfy the triangle inequality ...
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### General structure of the proof that every compact metric space is the continuous image of the Cantor set

I am reading the book "General Topology" by Stephen Willard and I have the theorem "Every compact metric space is a continuous image of the Cantor set". The book also presents its proof. The proof ...
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### How do you find the metric tensor for a given manifold?

Is there some general way to derive the metric tensor for a given manifold M? For example, how was the metric for the surface of a sphere $$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$ first derived?
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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 ...
### $X$ 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 ...
### $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ 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 ...