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 a set is compact - Homework

Let $(X,d)$ be a metric space and let {$p_n$} be a sequence of points in $X$ with $\lim_{n\to ∞}p_n = p_0$. Prove that the set $K =$ {$p_0, p_1, p_2,...$} is a compact subset of $X$. I have ...
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Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true?

Is $\frac{\vert x+y\vert}{1+\vert x+y\vert}\leq\frac{\vert x\vert}{1+\vert x\vert}+\frac{\vert y\vert}{1+\vert y\vert}$ true for all $x,y\in\mathbb{R}$? If not, how can I prove that $\int\frac{\vert ...
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Closed Interval in $E^2$

I am currently working through Introduction to Analysis by Rosenlicht In one of the exercises $4.30,$ he asks a question regarding a closed interval in $E^2.$ I am not sure what this means. I was ...
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50 views

Any open set shares boundary with a discrete set

Claim: Let $X$ be a metric space and let $U\subset X$ be open. Then there exists a discrete set $A\subset X$ such that $\partial A = \partial U$. Approach thus far: Since this statement is about ...
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86 views

Find sets of points, where function from one topological space to another is continuous.

We have got two functions : $f(x,y) = (2x,y)$ $g(x,y) = (x+1,y) $ They are transormations from one topological space to another ( from $ (\mathbb{R^2}, \tau')$ to $ (\mathbb{R^2}, \tau'')$ ), ...
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75 views

Necessity in Arzela-Ascoli theorem

I am trying to prove necessity of boundedness and equicontinuity in Arzela-Ascoli and I don't know how to go about it. More precisely,I have: Let $K$ be a compact metric space, and $A\subset C^0(K)$ ...
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Having difficulties showing the triangle inequality of metric in the plane

Let $P \in \mathbb{R}^2$ and define $$ d(x,y) = \left\{ \begin{array}{lr} ||x-y|| & \text{if} \; \; x,y,P \; \; \text{are collinear,}\\ || x - P|| + ||y-P||& \;\;\;\; ...
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Distance function is in fact a metric

I know I should be able to show this, but for some reason I am having trouble. I need to show that $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ is a metric on $\Bbb R$ where $|*|$ is the absolute value metric. ...
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How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
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35 views

Disjoint open sets in $\mathbb{R}^N$

I have a set exercise which says: Prove that in $\mathbb{R}^N$ with the Euclidean metric any collection of disjoint open sets is at most countable. Is this true for any arbitrary metric space? Now I ...
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Continuous function with infimum

Let $A$ be a closed subset of a metric space $X$ and $f:A \rightarrow[1,2]$ a continuous function on it. Now I want to find out why the function $$F(t):=\frac{\inf\{f(s)d(s,t);s \in A\}}{\inf ...
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56 views

Closed sets in a subspace are formed by intersecting the subspace with closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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1answer
29 views

Distance to a set

I have a question concerning to the following problem. Let $(X,d)$ be a metric set. For every subset $T \subset X$ we define a mapping \begin{equation} d_T : X \rightarrow R , d_T(x) := inf\{d(x,y) | ...
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1answer
61 views

Existence of maximizer implies compact? [duplicate]

I know that compact sets imply the existence of a maximizer, but is the converse true: Let $(X,d)$ be a metric space. Suppose that whenever $f$ is a continuous (and real) function on $X$, there ...
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MAth proof questions Open closed sets

Let $X$ be a metric space and let $Y$ be a subset of $X$ be a subspace with the induced metric. (induced means the metric restricted to elements of $Y$) Let $A$ be a subset of $Y$. Prove the following ...
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79 views

properties of connected set

If $C $ is a connected set in a metric space $X$ & $C$ intersects both $A$ and $X\cap A^c (A\subseteq X)$ then can it be concluded that $C\cap \delta A\neq\phi$ ?
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Is $SO(2,\mathbb R)$ connected

Is $SO(2,\mathbb R)=\{ A\in O(2,\mathbb R): det A=1\}$ connected?Why?
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120 views

Topology-Open Sets of a Metric Space

Let $(X_i,d_i), i=1,2,\dots,n$ be metric spaces. Let $X=\prod_{i=1}^{n}X_i$ and let $(X,d)$ be the metric space defined in the standard manner. For $i=1,2,\dots,n$, let $O_i$ be an open subset of ...
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Question about Compact metric space

Please why $$\Delta_p=\lbrace (t_0,...,t_p)\in \mathbb{R}^{p+1},t_i\geq 0, \sum_{i=0}^p t_i=1\rbrace$$ is a compact metric space ? Thank you
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showing completeness of a metric space

$X=\mathbb{R}_{>0}$, $d(a_1,a_2)=|\ln(a_1)-\ln(a_2)|$. I have already proven that $(X,d)$ is a metric space, but I have some problems showing the completeness. Let $(a_n)_{n\in\mathbb{N}}$ a ...
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What is the maximal size of an equal-distance set in $\mathbb{R}^n$?

Let $A\subseteq \mathbb{R}^n$ with the casual metric and $c\in\mathbb{R}^+$ be a real positive number, such that for every $a_1, a_2\in A$ if $a_1\neq a_2$ then $d(a_1,a_2)=c$. What is the maximal ...
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Why open unit ball in any infinite dimensional Banach space is finitely chainable?

In paper "Pointwise products of uniformly continuous functions" by Sam B. Nadler, Jr., He defined the finitely chainable as followings : Let $(X,d)$ be a metric space. An $\varepsilon$-chain in ...
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Approximation of acontinuous function

How to approximate a continuous function on $[-\pi,+\pi]$ which is $2\pi$ periodic by a set of trigonometric polynomials in the sup-norm topology?
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is the set of matrices with trace equal equal to zero compact

Is it true that the set of all matrices with trace equal to zero a connected and compact subset of the 2*2 matrices over R?
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67 views

Metric Space and Open Sets

I'm having trouble figuring out where to go with this problem. Any hints or strategies would be appreciated. I have just the basic definitions for open sets, distance metrics, etc. Consider $\Bbb ...
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1answer
21 views

Continuity of function and its value.

Here's a problem I'm struggling with. Not really sure how to do this. My tools are epsilon delta proofs for continuity and that's about it. Let $f:[0,\infty)\to\Bbb R$ be a function which is ...
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48 views

New metric spaces from given one

Let (X,d) be a metric space, $f : [0,\infty) \rightarrow [0,\infty)$ continuous differentiable, strict monotone increasingly with $f(0)=0$ and a monotone decreasingly derivative. Prove that $f \circ ...
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74 views

Translation invariant metrics and topological groups

During a lecture was pointed out that one of the main feature, from a topological perspective, of normed vector spaces is the translational invariance, that is that one can study the topological ...
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63 views

Hausdorff Distance between Subdifferential sets

Suppose I have two convex functions $f$ and $g$ mapping $\mathbb{R}^{n}\to\mathbb{R}$ (so they are 'more' than proper). Suppose $\|f-g\|_{\infty}<\epsilon$, the sup-norm. In particular, the ...
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Metric Space Proof - Analysis

Let $\mathcal{C}([0, 1])$ be the set of all continuous functions $f : [0, 1] \to \mathbb{R}$. For $f, g \in \mathcal{C}([0, 1])$, define $d(f, g) = \max|f(x) − g(x)|$. Show that $d$ is a metric on ...
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Proof of that there is no metric on $\mathbb{R}$ which is equivalent to the natural metric and which induces a metric on $(0,1)$

I want to prove the following statements: For $X:=(0,1),$ prove the following: (a) $d(x,y):=\big|(1/x)-(1/y)\big|$ is a metric on $X.$ (b) The natural metric and $d$ are equivalent. (c) There is ...
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Homeomorphic metric spaces

I want to examine if $(0,1] $ and $\mathbb R $ are homeomorphic. We work on metric space $(\mathbb R, e)$, where $e$ stands for the euclidean metric. My answer: Let's assume there is a ...
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Norm on $\mathbb R^n$ with given unit ball

Consider a finite subset $S$ of $\mathbb Z^n$ such that $-s\in S$ whenever $s\in S$ and $S$ generates $\mathbb Z^n$. What is a norm on $\mathbb R^n$ whose unit ball is precisely the convex hull of ...
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104 views

Show that it is at most countable

In a space of finite measure, show that a family of disjoint measurable sets with positive measure is at most countable. Could you give me some hints what I am supposed to do??
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Continuity of a function?

Let $f: (\mathbb N, e_\mathbb N) \to(\mathbb R,e)$, where $e$ stands for the euclidean metric and $$f(n)=\begin{cases} n,\, n\ge 2\\\\0,\, n=1\end{cases}$$ Is $f$ continuous? Firstly, I can ...
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Some Strange Minimum and proof

I read following on Norm chapter in one book. $$\begin{align}\left|\left\|x-y\right\|-\left\|w-z\right\|\right| \leq & \min \{\|x-w\| + \|y-z\| , \|x-z\| + \|y-w\|\}\\\text{ or, }&\min ...
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Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
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Topological Vector Space not induced by Metric

Can anyone give me an example of a Topological Vector Space that is not metrizable? I know that the neighborhood base of $0$ needs to be incountable, and all I can construct then is no topological ...
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continuous function from one metric space to another metric space

Is differentiation $f(x) \rightarrow f'(x)$ a continuous function from $C^1[a,b] \rightarrow C[a,b]$ ? Is integration $f(x) \rightarrow \int_a^x \! f(t) \, \mathrm{d}t $ a continuous function from ...
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86 views

Topologies generated by a metric

Hi I am new to mathematical proofs/notation and am working through John Lee's Introduction to Topilogical Manifolds. This is the question and my attempt. This is not homework. 2.4 Suppose $M$ is a ...
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topological properties of a given set

Let us consider the set $X=C[0,1]$ with its sup-norm topology. Let $S $ be the set of all elements $f$ of $X$ such that $\int_0^1 f(t) dt=0$. Is $S $ compact and connected? To show $S$ compact I have ...
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Density of sets

I have got a problem on whether a set is dense or not but not quite sure on how to approach it. Consider the space $M_2(R)$ with its usual topology.Consider the set $ S$ of all matrices with both ...
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Show that there is a series in R^infinity has some term greater than or equal to 1/n but that also is arbitrary close to the zero sequence.

Consider $\mathbb{R}^\infty = \{(a_n): \sum_{n = 1}^{\infty} a_n^2 < \infty\}$ with the metric $d((a_n$), ($b_n$)) = $[\sum_{n=1}^{\infty} (a_n - b_n)^2]^{1/2}$. Let $A = \{(a_n) : |a_n| < 1/n ...
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Closed or open subsets of $C[a,b]$? [duplicate]

$C[a,b]$ denotes the space of continuous real-valued functions on $[a,b]$. The metric associated with $C[a,b]$ here is $d(f,g)=sup[|f(x)-g(x)|]$ where the supremum is taken over $[a,b]$. $C^1[a,b]$ ...
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Why isn't $\,\mathcal C[0,1]$ a Banach space in this unusual norm?

I wish to ask the following question: Let $\mathcal X$ be the normed space $\,\mathcal X=\mathcal C([0,1])$, with norm defined as $$ \|\,f\|= \max_{x\in[0,1]} x^2 \lvert\,f(x)\rvert. $$ Why isn't ...
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limit in a metric space

Let $(X, d)$ and $(X, d_1)$ be two metric spaces over the same set $X$. Suppose that a sequence $(a_n)$ in $X$ converges in $(X, d)$ to $l$ and converges in $(X, d_1)$ to $l_1$. Then must $l$ equal ...
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In a semi-metric space, need the limit of a sequence be unique as it is in a metric space? Yes

In a metric space (M,d), define what you mean by a bounded set B and by L being the limit of a sequence $\{x_n\} \in M$. What is the sequential definition of a function between two metric spaces being ...
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Is the set $E=\{0.a_1a_2… \in \mathbb{R}\mid a_i= 4 \text{ or } a_i=7\}$ dense, compact or perfect?

I want to check my reasoning, I found that it's not dense but it's compact and perfect. $1$- It's not dense for 1 is neither in the set of a limit point of it. $2$- It's compact because it's both ...
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Give 3 different examples of semi-metric spaces which are NOT metric spaces.

A semi-metric space (M,d) satisfies all of the conditions of a metric space except it need NOT satisfy $d(f,g)=0 \iff f=g$. Give 3 different examples of semi-metric spaces which are NOT metric ...
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Why $\mathbb R$ is not complete with the metric $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$?

Suppose $d(x,y)=|\phi(x)-\phi(y)|$ where $\phi(x)=\frac{x}{1+|x|}$. Prove that $\mathbb R$ is not complete with this metric. This is exercise 12 from chapter 1 from Rudin's Functional Analysis. ...