# Tagged Questions

Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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### Why is this set closed?[metric-spaces]

I am reading a note, where part of it is this: Why is S' closed? I have tried to argument like this, but I am not able to finish the argument: Let $\{x_n\}$ be a convergent sequence in S', then ...
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### Prove the equation $\vert d(x,z)-d(y,t)\vert\leq d(x,y)+d(z,t)$

I know how it verified the following equation: $$\vert d(x,y)-d(x,z)\vert\leq d(y,z)$$ where $x,y,z$ is arbitrary points of metric space $(X, d)$ But I didn't now how to prove the follow equation: ...
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### How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
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### Isometry fixing an open set pointwise is identity

Let $X$ be a metric space and $F$ an isometry of $X$. Suppose $F$ fixes each point of a non-empty open set $U\subset X$. Under what conditions on $X$ does it always follow that $F=\mathrm{id}$? I ...
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### The future of the orbit of a point is a closed set [duplicate]

$X$ is a metric space and $f: X \rightarrow X$ is a dynamical system. Prove: $w(x_{0})$ is closed. Here the set $w(x_{0})$ is the future of the orbit of $x_0$, defined as \omega(x_0) = \{y \mid ...
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### How are delta distributions defined in metric spaces?

How are delta distributions defined in metric spaces with continuous metric?
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### Metric spaces not isometric to any of their proper subsets

Let's say a metric space $X$ has property $P$ if $X$ is not isometric to any of its proper subsets. I'd like to know what this property is called in the literature and whether there's a nice ...
### For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$
Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...