# 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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### How to compute antilogarithmic and superlogarithmic spaced values?

Let's suppose I have a range, e.g. $[100, 900]$. I want to compute 8 logarithmic spaced values $x_i={100, ..., 900}$. I use the following formula: $$x_1=\log(S)+\frac{(i-1)\log(S/L)}{n-1}$$ In the ...
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### Simplexes in $\mathbb R^n$ have at most $n+1$ points

This is an exercise from the book Espaços Métricos (metric spaces) by Elon Lima. I'm translating it (the part of it that I'm having trouble with): Show that if $X\subset\mathbb R^n$ is such that ...
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### Suppose $f \in B_{d_1}(g,\epsilon)$ can we conclude that $f \in B_{d_2}(g,\epsilon)$

Suppose $f \in B_{d_1}(g,\epsilon)$ can we conclude that $f \in B_{d_2}(g,\epsilon)$ where the space is $C[0,1]$ and $d_1$ is the metric induced by the $1$ norm and $d_2$ is the metric induced by the ...
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### balls have empty boundary with regard to the $p$-adic norm

Let $p$ be prime, $a\in\mathbb{Q}$ and $r\geq0$. How can I show that the closed ball $D(a,r)$ in $(\mathbb{Q},|\cdot|_p)$ must have an empty boundary (with regard to the topology induced by the ...
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### Is $T:(x,y)\mapsto(x+\alpha, y+x)$ mod $1$, expansive on $\mathbb{R}^2 / \mathbb{Z}^2$?

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha, x+y\right) \mod 1$$ One ...
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### For every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then to prove $f$ is a polynomial in $[a,b]$

Let $f:[a,b] \to \mathbb R$ be a continuous function having derivatives of all order such that for every $x \in [a,b] , \exists n_x\in \mathbb Z^+$ such that $f^{(n_x) }(x)=0$ ; then how do I show ...
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### To Prove that $(X,d)$ is not separable.

Let $X$ be the set of sequences in $[-1,1]$. Define $d$ on $X\times X$ to be $d(a,b) := \sup\{|a_n -b_n| :n \in \mathbb N\}$. Then $d$ is a metric on $X$; and show that $(X,d)$ is not separable. I am ...
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### If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
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### Proving intersection of dense subsets of a metric space X is the isolated points of X.

Suppose X is a metric space. Let $\mathscr C$ denote the collection of all dense subsets of X. Show that $\bigcap\mathscr C$ = iso(X). Thus the question asks to prove that every dense subset of X ...
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### Property of distance and adherence

Please how to prove that in a metric space $(E,d),$ for $A,B\subseteq E$ that $\forall x\in E, d(x,A)=d(x,\overline{A})$ and that $d(A,B)=d(A,\overline{B})=d(\overline{A},\overline{B})$ and ...
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### Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
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### Equivalence of Theorems; the sphere on $\ell^2$ is finitely oscillation stable.

I just started reading the book Dynamics of Infinite-dimensional Groups, by Pestov, and right in the introduction the following theorem by Milman is cited: Let $\mathbb{S}^\infty$ denote the sphere in ...
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### Gradient of the distance function

Let $\Omega$ be open, bounded subset of $\mathbb{R}^n$. Let $d(x):=dist(x,\partial\Omega)$ denotes the distance of the point $x\in\Omega$ from the boundary $\partial\Omega$. Define function ...
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### Totally disconnected vs. totally separated.

Assume all spaces are metric. Question. Does there exist a space $X$ which is totally disconnected (the components of $X$ are singletons), yet some quasicomponent of $X$ has nonempty interior? I ...
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### Elements with infinite roots in p-adic

Let $\mathbb{Q}_p$ the $p$-adic completion of $\mathbb{Q}$ and $$S=\{x\in\mathbb{Q}_p:1+x\mbox{ has n-th root in }\mathbb{Q}_p\mbox{ for infinite }n\in\mathbb{N}\}$$ I have to show that ...
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### A not complete metric space?

Please ,how to prove that the space $\mathbb{R}$ endowed with the metric $d(x,y)=|e^x-e^y|$ is not a complete space? I don't find a Cauchy sequence but not convergent Please Thank you.
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### How is a metric space a topological space? [duplicate]

I learned about metric spaces and topological spaces but I don't see how they correlate. How does a metric space follow the properties of a topological space.
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### Prove that a set in a metric space cannot be both open and closed.

If I have a metric space $X$, and $E \subset X, E \ne X, E \ne\emptyset$. I want to prove that E cannot be both open and closed. I have two strategies, but I am not able to finish them: I assume ...
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### Convergence of distances in metric space

If $(X,d)$ is a metric space, $(x_n)$ and $(y_n)$ are Cauchy sequences in $(X,d)$. How do i show that $(a_n):=d(x_n,y_n)$ converges? Here is what i did: Let $(x_n)$ and $(y_n)$ be Cauchy sequences, ...
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### Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
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### Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
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### Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$? [closed]

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
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### Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...
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### Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
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### Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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### How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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### A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
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### Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
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### A geodesic metric space is a manifold on its own right. What are conditions for a Finsler space to be a manifold?

A geodesic metric space can locally be approximated by a vector space. This approximation provides it with a natural manifold structure. It means that geodesic metric space is more fundamental concept ...
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### Is this a metric on matrices?

In the set of $n$-by-$n$ reversible real matrices, decide whether $$d(A,B)=\ln (\lVert A^{-1}B\rVert\cdot\lVert B^{-1}A\rVert)$$ defines a metric and/or semi-metric. Can you please help me to solve ...
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### Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
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### Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric ...
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### concepts which is present in metric space but not in topological space

I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind ...
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### A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
### Prove ${x:d(x,p) < d(x,q)}$ is open in metric space $X$
$X$ is a metric space and $p \neq q$ $\in X$. I want to prove that $E=$ $\{x:d(x,p) < d(x,q) \}$ is open in metric space $X$. I think I can directly prove this by showing every point $x \in E$ ...
I am studying metric spaces and got confused about many different ways of defining the closure. Let $S$ be a subset of $M.$ Then, the closure of $S$ is \$ \{x \in M : \forall \epsilon>0, \ \ ...