# 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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### I've got a definition, but says something strange, what does it mean?

$M_1:=(M,d_1), \ \ M_2:=(M,d_2)$. $d_1$ is equivalent to $d_2$ if the identity $x\rightarrow x$ of $M_1$ over $M_2$ is an homeomorphism I'm not sure what it is talkin about when it says "identity" ...
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### Help understanding a proof that the metric space of bounded functions is complete.

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
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### Why does the second statement follow from the first one?

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
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### Showing that the given interval is is connected.

Let $(R,d)$ be the space of real numbers with the usual metric. Let $I$ be an interval such that $I \subseteq R$. We need to show that $I$ is connected. Let us say that $I$ is not connected , thus ...
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### Does every connected metric space , with more than one point , contains a path connected subset with more than one point ?

Does every connected metric space , with more than one point , contains a path connected subset with more than one point ? Is there any additional condition imposing which on the mother space will ...
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### Let $I = [0, 1) ∪ [2, 3]$ with the usual metric.Prove that the closure of $B$ is not the closed ball of center $0$, radius $2$.

Let $I = [0, 1) ∪ [2, 3]$ with the usual metric. Let $B = I(0, 2)$ be the open ball of center $x$ and radius $1$. Prove that the closure of $B$ is not the closed ball of center $0$, radius $2$. The ...
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### Difference between a continuous function and an isometry? Is a continuous function a homomorphism?

Definition of continuous function on a set: Suppose $X$ and $Y$ are metric spaces. Let $f: X \to Y$. We say $f$ is continuous on $X$ if for every $\varepsilon >0$, $\exists \delta >0$ such that ...
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### Fixed Point with two equivalent metric complete spaces

Can anyone help me with this: Let $(X,d)$ and $(Y,d')$ be two metric spaces. We know that if the metrics $d$ and $d'$ are equivalent (strongly, that is), the completeness of one implies that of the ...
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### Contraction mapping theorem for metric spaces - why is this part of the proof notable?

Someone has made a remark to me that from the proof of the CMT for metric spaces we get that $$d(x, x_n) \leq \frac{K}{1-K}d(x_{n-1}, x_n).$$ What is this saying? Why is this remark worthy? (The ...
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### What is “the set in which this* satisfies lipschitz condition”?

I'm very confused, I believe my teacher was not clear enough, maybe im missing something. $$1)y''+2ay'+by=0$$ In my "definition of lipschitz condition": G is the domain of a function f in which you ...
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### Prove $(\Bbb R^{n},d_{p})$ forms a complete metric space

Prove $(\Bbb R^{n},d_{p})$ forms a complete metric space So I know that in order for a metric space to be complete I must show that every Cauchy sequence in $\Bbb R^{n}$ converges on $\Bbb R^{n}$ ...
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### Is the set of polynomial dense in $C[-1, 0]$?

If the set is defined as $$\{a_0+a_1x+a_2x^2+...+a_nx^n, \text{where }n\ge 0 \text{ and } a_0+a_1+...+a_n=0\},$$ is the set dense in $C[0,1]$ and $C[-1,0]$? For the first question, I'm thinking ...
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### Isometries of metric spaces $Z=X\cup (X\times\mathbb{R})$ (corrected)

Let $(X,d_1)$ be a compact metric space and consider $Z=X\cup (X\times\mathbb{R})$. Consider the metric $d$ on $Z$ defined by: $$d(x_1,x_2)=d_1(x_1,x_2)$$ $$d(x_1,(x_2,t_2))=d_1(x_1,x_2)+1$$ ...
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### Arzela-Ascoli Theorem on metric spaces

I've been looking for a proof of one particular direction of this theorem for metric spaces. I've looked online, but everyone seems to use different terminology/notation to state the theorem, so I'd ...
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### open ball on metric $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$ in $\mathbb{R}^2$

I have the following metric: $d''(z,z') = \max \{d_i(x_i,x_i'), i\in \{1,\cdots,n\}\}$ in $\mathbb{R}^2$ What figures form the open and closed ball? What about the sphere? That's what I thought: ...
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### If $X$ is a countable dense subset of the separable metric space $P$, then $P \setminus X$ is not separable [closed]

If the metric space $P$ is separable, then there exists a countable dense subset $X\subset P$.but if P is uncountable, Prove that $P \setminus X$ is not separable.
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### $X,Y$ metric spaces , $X$ complete , $Z$ is Hausdorff , $f,g:X \times Y \to Z$ continuous in each variable and coincide on a dense subset , is $f=g$?

Let $X,Y$ be metric spaces , $X$ be complete and $Z$ be a Hausdorff space ; let $f,g:X \times Y \to Z$ be functions such that each of $f$ and $g$ is continuous in $x \in X$ for each fixed $y \in Y$ ...
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### A notion of nonpositive curvature for general metric spaces

The proof of the following result should be done by using the second variation formula of geodesics but I do not know how to start or what is the main idea of the proof. (Lemma 3.7 in the paper: A ...
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### Prove that $\mathbb{R}$ is not isometric to any proper subset of $\mathbb{R}$.

The question is in the title. I can show easily that $\mathbb{R}$ is not isometric to any closed interval $[a,b]$ in $\mathbb{R}$, but I am struggling to show it for any arbitrary subset.
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### Completeness and Compactness of Cartesian Product of Metric Spaces

Suppose $\{(X_i,d_i)\}_{i\in \mathbb{N}}$ is a collection of metric spaces with the cartesian product defined by $A=\prod_{i=1}^{\infty}X_i$. Let the metric on $A$ be given by ...
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### Random Geometric Graph in unit disk

According to the Wikipedia article, to define a random geometric graph, one needs a metric space. In the examples they give for 2D random geometric graphs, they say that "an RGG can be constructed ...
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### I'm not certain this makes any sense: Matrix Multiplication of Metric Tensor for calculating arclength

I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Arclength Where in it gives the euclidean measure of distance as $$ds^2 = E du^2 + 2 F du dv + G dv^2$$ Equivalently as  ds^2 ...
### Is it true that any continuous function $f$ on $[0,\infty)$ can be approximated by polynomials?
I think it's true, but how to prove it? By weierstrass approximation theorem we can approximate uniformly any continuous function on a closed interval $[a,b]$ by a sequence of polynomial function ...
### If $f$ is not uniformly continuous function on $(0,1)$ then $f$ can't be extended to a continuous function on $[0,1]$.
Is this proposition correct? If it is correct can anyone prove it? Because I need it to prove that if $f$ is a continuous function (not uniformly continuous) on $(0,1)$ then it can't be approximated ...