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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What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
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Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
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A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
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Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
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subset of a complete space has a relatively compact $\varepsilon$-net.

Trying to prove that, a subset $A \subset X$ of a complete space $X$ is relatively compact iff $\forall \epsilon > 0$ $A$ has a relatively compact $\epsilon$- net. I have proved the following ...
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Definition of topological space

The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the ...
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Can we define the derivative of a function in arbitrary metric space in the following way?

Let us first define some terms. Definition of Pre-pseudometric Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if, ...
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Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
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How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
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Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
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Is there a term for this line like subset within a metric space

While thinking about geodesic lines I started exploring subsets of a metric space that have the following property. $ \forall a,b,c \in L, d(a,c)> d(a,b) \land d(a,c) > d(b,c) \implies d(a,c) = ...
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If $Z(f)$ is the zero set, prove that $Z(f)$ is closed

Introduction: Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98. Exercise: Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of ...
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Is the empty set an open ball in a metric space?

Problem Let $(X,d)$ be a metric space where $X$ is a non-empty set. Is the empty set an open ball in $X$? I think that it is true because if $X=\mathbb{R}$ with the usual metric then for all ...
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proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$

Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove. Is there any simple proof (or counter-example) for the proposition: Consider the closed ball of ...
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EDITTED: Find all values of $a$ and $b$ so that $ax^n+b\cos\left(\frac{x}{n}\right)$ is Cauchy.

For each $n\in\mathbb{N}$ let $$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$ Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of ...
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Any bi-invariant distance on a group is inverse-invariant?

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
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Is every compact metric space finite dimensional?

It certainly is true if the metric space is normed by Riesz lemma.
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Equivalent Metrics on $\mathbb{R^n}$

I am working on a problem and want to verify that my logic and reasoning is correct. This is my first time working with metric spaces. Show that the following define equivalent metrics on ...
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Does a mapping from one metric space to another metric space preserve star-likeness of regions?

Let $X$ be a vector space, let $M_1 = \left(X, d_1\right)$ be a metric space and let $M_2 = \left(X, d_2\right)$ be another one. $f : M_1 \to M_2$ is continuous and the origin is a fixed point. $f$ ...
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Is a metric space a requirements for the application of the algebra of events from probability?

When I refer to a metric space, I mean a space that has some genuine notion of distance. In some applied context, this distance would be computed with respect to a coordinate system. I just wanted to ...
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What is the connection between $l_p$ norms and “$l_p$ metrics”?

In some textbooks metric spaces you sometimes encounter these "$l_p$ metrics", $d_1, d_2, d_\infty$ (I don't think $l_p$ metric is very standard usage) For example, $d_1(x,y) := \sum\limits_i^m ...
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How to perform statistical test for two sets of points?

(I have asked this question originally on Cross Validated; however, no good answer and someone suggested me to ask the question here). Thanks a lot in advance if anyone can help. We know that we can ...
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Grassmannian Non-Convex

The Grassmannian manifold $Gr(r,V)$ defines the set of $r$-dimensional linear subspaces of the vector space $V$. My question is, in general, what is the simplest way to see that $Gr(r,V)$ is a ...
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Showing interior of a set is empty

Consider the metric space $(C[0,1],d_{\infty})$. For $x_{0} \in [0,1]$ and $M > 0$ define the set $A \subset C[0,1]$ by $$ A = \{f \in C[0,1] \phantom{.}|\phantom{.} |f(x) - f(x_{0})| \leqslant ...
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Trouble understand a step in the proof that $l^p$ is complete

I'm reading through a proof, attached here. I didn't include the whole proof. The last step is the one I'm confused about. Shouldn't there be more of a justication for taking $\lim_{n \to \infty}$ ...
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1answer
21 views

Distance function is continuous in topology induced by the metric

The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ ...
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Is this set open in the product topology?

Let $X$ and $Y$ be topological spaces and equipp $X\times Y$ with the product topology. Assume $U\subset X$ is open and for every $p\in U$ we have an open subset $V_p\subset Y$ of $Y$. Is the set ...
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Product spaces and open sets

I have a proposition I have been pondering that I need help with. Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. Recall that the product space $(X\times Y, d_{1})$ is also a metric space with the ...
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Creating a configuration of points where each point is away from all other points by a pre-defined distance

Let's assume that the points $\in \mathbb{R}^2$ and there are only C=5 points (in practice, I may have $\mathbb{R}^{800}$ and 1000 points). The first out of the five points is fixed. We also have been ...
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Topology - Open set [closed]

How do I show this? Let $(X,d)$ be a metric space and $x\in X$ an element in $X$. Show that \begin{align*} \{y\in X|d(y,x)>r\} \end{align*} is open for any $r\in \mathbb{R}$. The definition of a ...
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Is it always possible to find a “pre-metric” from a metric?

Problem Let $X$ be a non-empty set. Let $f:X\times X\to \mathbb{R}$ satisfying the following properties, $f(x,y)=0\iff x=y$ for all $x,y\in X$. $f(x,y)=-f(y,x)$ for all $x,y\in X$. ...
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Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
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Theorem 2.41 in Baby Rudin: Is this proof good enough? Can we generalise it?

Here is Theorem 2.41 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If a set $E$ in $\mathbb{R}^k$ has one of the following three properties, then it has the other ...
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Are sequences themselves metric spaces with the inherited metric?

I have been asked to show whether a sequence $(p_n)$ in $\mathbb{R}$ is a metric space with the inherited metric $d(x,y) = |x-y|$ It seemed to me at first to be a slightly odd question because we ...
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If two nested open sets have the same nonempty boundary, are they the same set?

Let $(X,d)$ be a metric space. Let $B_\epsilon(x)$ be the open ball of radius $\epsilon$ centered at $x$. For $x\in X$ and $\epsilon>0$, suppose that $V$ is an open set in $X$ with $V\subseteq ...
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Subset of $(l^{2},d_{2})$ is open

Show that $A = \{\phantom{i}\{x_{n}\} \in l^{2} \hspace{2mm}:\hspace{2mm} |x_{n}| < 1, \forall \phantom{i}n \in \mathbb{N}\phantom{i} \}$ is open in $(l^{2},d_{2})$. The $d_{2}$ metric is: $$ ...
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Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...
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Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense ...
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Why is it that $\inf_{x∈E} d(x, a) > 0$?

Suppose $X$ is a metric space, $E ⊂ X$ is closed, and $a$ is a point not in $E$. Why is it that $$\inf_{x∈E} d(x, a) > 0$$ ?
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Equivalent conditions for continuity in a metric space

Let $f: (X,d_{X}) \to (Y,d_{Y})$. We have that $f$ is continuous iff for any open subset $U \subset Y$, $f^{-1}(U)$ is open in $X$. Prove the following are also equivalent to the definition for the ...
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Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
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Limit Points of closure of A is subset of limit points of A

For a metric space $(M,d)$ and a subset $A \subset M$, is it true that the set of limit points of the closure of $A$ is a subset of the limit points of $A$? ( I have managed to prove the reverse ...
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If $|X|<\infty$ then T metrisable $\rightarrow$ T discrete topology.

If $|X|<\infty$ then T metrisable $\rightarrow$ T discrete topology. I said let $d$ be a metric and let $x \in X$. I want to show that $\{x\}$ is open. How do I show this?
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How can we find a contradiction?

Let $(X, \rho)$ be a metric space and $x \in X, A \subset X (A \neq \varnothing)$. We have $x \in \overline{A}$ iff $d(x,A)=0$. We suppose that $d(x,A)=0$ . We want to show that $x \in ...
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how do I prove the following set is open?

let $(X,d)$ be a metric space and let $A$ and $B$ be two disjoint closed sets in $(X,d)$. define $U=\{x\in X | d(x,A)<d(x,B)\}$. how do I prove that $U$ is open in $(X,d)$?
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Show a unit disk with max metric is closed?

I must show that if $\mathbb R$2 is equipped with max metric, d = (($x$1, $x$2),($y$1, $y$2)) = max{|$x$1 - $y$1| , |$x$2 - $y$2|} then the disk D = {($x$1, $x$2) ∈ $\mathbb R$2 : $x$12 + $x$22 ...
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Interior points: Precise definition

Let $(M,d)$ be a metric space. Then $ x \in S $ is an interior point of $ S $ if some ball centered around S of positive radius is wholly contained in $S$. But consider this. The set $S_{L}$ of all ...
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Why is $U ⊂ \mathbb{R}^n$ open with respect to metric $d_p$ iff it is open with respect to metric $d_q$ for $q ∈ [1, ∞)$?

Let's say that for any $p ∈ [1, ∞)$ we have a distance function on $\mathbb{R}^n$ given by $$d_p(x, y) := \left(\sum^n_{j=1}|x_i - y_i|^p\right)^{\frac{1}{p}}$$ How would I show that a set $U ⊂ ...
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Single reference to classical results in analysis.

I am writing an expository work. And I need classical references (books or articles) that simultaneously proof the three classical results below. Any suggestion? Theorem. Let ...
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Name for metric with property that each point is separated…

Let $(X,d)$ be a metric space such that for each point $x \in X$ we have the property that $\inf\{d(x,y) \,\vert\,x\neq y,\,y \in X\} > 0$. i.e each point can be separated by some small ball. Is ...