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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Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
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1answer
132 views

Fréchet mean between points in $\mathbb{R}^3$

Let $X$ be a set of $n$ points in $\mathbb{R}^3$ and $f_m$ be the Fréchet mean, i.e.: $$ f_m= \arg \min_{p \in M} \sum_{i=1}^n w_id^2(p,x_i) $$ where $(\mathbb{R}^3,d)$ is a complete metric space, ...
6
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1answer
112 views

About a continuous function

I'm trying to solve this problem, but I don't have any idea. Can you help me? Let X a compact metric space and $f:X\times\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function. Consider ...
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3answers
678 views

M compact $p\in M$ , there exist $f:M-p\to M-p$ continuous bijection but not homeomorphism?

Let M be a compact metric space. We know that if $ g:M\to M$ is a continuous bijection then it's a homeomorphism. But I want to know, if I have a continuous bijection $ f:M - \left\{ p \right\} \to M ...
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1answer
631 views

Continuous function on metric space

I'm trying to show: Let $(X,d)$ be a metric space and let $A, B$ be nonempty subsets, which are also closed and disjoint. Let $\rho_A:X\to \mathbb{R}$ be such that $\rho_A=d(x,A)$ and $\rho_B:X\to ...
1
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0answers
162 views

Checking for completeness of $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$

I have to check for completeness of following metric spaces 1 : $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$ for all $x, y \in \mathbb{R}$. 2: $\mathbb{Q}$ with metric ...
4
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1answer
162 views

If $d_1, d_2$ are metrics of $X$, is it true that $d_1 +d_2 $, $d_1 - d_2$, $d_1\cdot d_2$, $\sqrt d_1$ are metrics on $X$?

If $d_1, d_2$ are metrics of $X$, is it true that $d_1 +d_2 $, $d_1 - d_2$, $d_1\cdot d_2$, $\sqrt d_1$ are metrics on $X$? Here is my attempt: If we take $d_1 = d_2 $ = standard metric on the ...
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1answer
71 views

preimage of a pointwise limit of continuous functions and a $G_\delta$

Let be a sequence of pointwise convergent, and continuous functions $f_n : M \to \mathbb {R}$ , where M is a metric space. Prove that $\forall c\in \mathbb R$ , the set $ f^{-1} ([c,\infty ))$ is ...
2
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1answer
475 views

Sequential characterization of closedness of the set

Set $F\subset X$ is closed if and only if for every sequence $\left\{x_n\right\}\subset F$, if $x\in X$ and $x_n\rightarrow x$ then $x\in F$. [EDIT]: $X=\mathbb{R}^n$ with the usual topology. ...
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2answers
144 views

Problem on completeness, compactnes and connectedness of given subsets of $\mathbb{R}$

Let $S, T\subseteq \mathbb{R}$ be given by \begin{align*} S &= \left\{x\in \mathbb{R}:2x^2\cos\frac{1}{x} = 1\right\} \\ W &= \left\{x\in \mathbb{R}:2x^2\cos\frac{1}{x} \leq ...
4
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1answer
67 views

A problem of checking completeness of some subsets of $\mathbb{R}$

Let $S$ and $W$ be subsets of $\mathbb{R}$, with the usual metric, \begin{align*} S &= \left\{\frac{1}{n} :n\in \mathbb{N}\right\}\cup\{0\} \\ W &= \left\{n+\frac{1}{n}: ...
2
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1answer
210 views

Connectedness and compactness problem

In $\mathbb{R}^3$ with the usual topology, let: \begin{align*} V &= \{(x, y, z)\in \mathbb{R}^3 : x^2+y^2+z^2 = 1,\,y\neq 0\} \\ W &= \{(x, y, z)\in\mathbb{R}^3 : y = 0\} \end{align*} I ...
2
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3answers
176 views

Check for compactness and connectedness of subspace P = $\{(x, y, z)\in \mathbb{R}^3 : z = x^2+y^2+1\}$

I have got this problem in my exam: Check for compactness and connectedness of subspace P = $\{(x, y, z)\in \mathbb{R}^3 : z = x^2+y^2+1\}$ I think given subspace P is closed and bounded subset of ...
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1answer
1k views

Understanding open covering definition of compactness

My question is related with the understanding of open covering definition of compactness which can be stated as: A metric space X is called compact iff each of its open cover has a finite subcover. ...
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1answer
308 views

Extending a homeomorphism of a subset of a space to a $G_\delta$ set

I am having trouble figuring out the following question (3.10 in Kechris, Classical Descriptive Set Theory): If $X$ is completely metrizable, and $A\subseteq X$ with $f:A\to A$ a homeomorphism, then ...
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1answer
259 views

Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It ...
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1answer
271 views

Proof that a set has the Lindelöf property in a metric space

I am having some problems with the proof of the following Theorem: "Let $E$ be a set in a metric space $\mathscr{X}$. Then $E$ has the Lindelöf property provided there exists a countable set $D$ ...
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1answer
511 views

Two Disjoint Compact sets

let $S$ and $T$ be two disjoint compact nonempty sets. Show that there are points $x_0$ in $S$ and a point $y_0$ in $T$ such that $|x-y| \geq|x_0 -y_0|$ whenever $x$ is in $S$ and $y$ is in $T$.
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2answers
322 views

What's the need of defining notion of distance using norm function in a metric space?

I have started studying normed spaces. I wonder what's the need of defining notion of distance using norm function. For example , we know that $\mathbb{R}$ is a metric space with respect to usual ...
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1answer
130 views

An exercise of metric spaces in a continuous space

Is my work correct? Could they give me another counterexample?
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2answers
227 views

Space $Y\subset C[0,1]$ consisting of all $x \in C[a,b]$ such that $x(a) = x(b)$ is complete.

This is problem from kreyszig functional analysis: I have to show that space $Y\subset C[a,b]$ consisting of all $x \in C[a,b]$ such that $x(a) = x(b)$ is complete. I am struggling with this ...
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1answer
451 views

Set of all positive integers with metric $d(m,n) = \mid \frac{1}{m} - \frac{1}{n} \mid$ is not complete.

I am reading kreyszig functional analysis book where I got this problem: Let $X$ be the set of all positive integers and $d(m,n) = \mid \frac{1}{m} - \frac{1}{n} \mid$. I have to show that $(X,d)$ ...
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2answers
1k views

Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space

I have to show that the real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space. Certainly, I have to search for a cauchy sequence of real numbers ...
2
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1answer
998 views

Showing $(C[0,1], d_1)$ is not a complete metric space

I am completely stuck on this problem: $C[0,1] = \{f: f\text{ is continuous function on } [0,1] \}$ with metric $d_1$ defined as follows: $d_1(f,g) = \int_{0}^{1} |f(x) - g(x)|dx $. Let the sequence ...
2
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1answer
77 views

$M=X\cup Y$. If $S\subset M$ is open in $S\cup X$ and open in $S\cup Y$ then $S$ is open in $M$.

Let $M=X\cup Y$ be a metric space. If $S\subset M$ is open in $S\cup X$ and open in $S\cup Y$ then $S$ is open in $M$. I can't do anything with this exercise. I think that is the hardest problem ...
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4answers
369 views

Question about equivalent metric spaces

I have studied that topologically equivalent metrics produce the same open and closed sets. They also produce same compact and connected subsets. Does it mean that topologically equivalent metrics ...
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1answer
198 views

Why metric space $\mathbb{R}$ with the standard metric cannot be written as a countable union of nowhere dense sets?

I am trying to figure out why the metric space $\mathbb{R}$ with the standard metric cannot be written as a countable union of nowhere dense sets. Then, another natural question is: Can we write ...
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1answer
1k views

Understanding equivalent metric spaces

I have studied following definitions of equivalent metric spaces. Two metrics on a set $X$ are said to be equivalent if and only if they induce the same topology on $X$. 1: Two metrices $d_1$ and ...
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1answer
356 views

Understanding isometric spaces

I have studied that an isometry is a distance-preserving map between metric spaces and two metric spaces $X$ and $Y$ are called isometric if there is a bijective isometry from X to Y. My questions ...
2
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5answers
969 views

Distance between the point $x$ and the sebset A of a metric space $X$

Given a subset $A\subset X$ of a metric space (X, d) and $x\in X$. The distance between the point x and the set A is the infimum of the distances between the point and those in the set: $$d(x,A) = ...
6
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1answer
153 views

A set that is open in any metric space that contains it

Let $X$ be a set with the following property: For all metric space $Y$ such that $X\subset Y$ we have that $X$ is an open set on $Y$. $X$ should be the empty set?
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1answer
813 views

A basic question on diameter of a metric space

A set S of real numbers is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval. While in a metric space a non-empty ...
3
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4answers
2k views

Understanding the definition of Cauchy sequence

My question is related with the definition of Cauchy sequence As we know that a sequence $(x_n)$ of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer ...
6
votes
2answers
74 views

Mappings from $\omega_1$

We've been studying various properties of $\omega_1$ (equipped with the order topology), and I recently came across these questions. Can anyone help? If $f$ maps $\omega_1$ onto a metric space $X$ ...
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3answers
94 views

A metric in $\mathbb{R}^2\setminus\{0\}$

I can't find a metric $\delta$ in $\mathbb{R}^2\setminus\{0\}$ such that be equivalent to euclidean metric, be equal to euclidean metric in the unitary circle and for all $r>0$ the set ...
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1answer
160 views

Extending $f: X\subset \mathbb{R}^m\to\mathbb{R}^n$ an isometric immersion.

Let $X\subset \mathbb{R}^m$ not empty and $f: X\to\mathbb{R}^n$ an isometric immersion. Prove that there exists an isometric immersion $\varphi: \mathbb{R}^m \to\mathbb{R}^n$ such that ...
4
votes
2answers
689 views

Open and closed balls in $C[a,b]$

Let $X$ be a non empty set and let $C[a,b]$ denote the set of all real or complex valued continuous functions on $X$ with a metric induced by the supremum norm. How to find open and closed balls in ...
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3answers
792 views

Compactness of the Grassmannian

Let $V$ be a finite-dimensional inner product space. For $0 \leq d \leq \text{dim}(V)$, define the Grassmannian $G(V, d)$ to be the set of all $d$-dimensional linear subspaces of $V$, equipped with ...
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1answer
37 views

If $a=b+ c$ and $E$ is the basis of $a$, Will $E$ be also basis for $b$ and $c$?

Suppose $a$ lies in the span of a set of independent vectors $E$. Now, if $a=b+c$, is it also the case that $b$ and $c$ lie in the span o the same set of vectors $E$? if the question is obscure, ...
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1answer
356 views

'Nested Intervals Theorem' in $\mathbb{R}^2$

Cantor's Nested Intervals Theorem can be stated as "If $\{[a_n,b_n]\}_{n=1}^\infty$ is a nested sequence of closed and bounded intervals, then $\cap_{n=1}^\infty [a_n,b_n]$ is not empty. If, in ...
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votes
2answers
162 views

If $f:(M,d )\to (M,p)$ is a homeomorphism, are $d,p$ equivalent?

Let $M$ be a set and $\delta$, $\rho$ metrics on $M$. If $f:(M,\delta)\to(M,\rho)$ is a homeomorphism, are $\delta$, $\rho$ equivalent metrics? Not necessarly $f=\operatorname{id}_M$ (since result is ...
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3answers
181 views

Showing that $(1,2)$ is an open set

I was reading an example and it said $(1,2)$ is an open set in $(\mathbb{R},|.|)$. It showed that $(1,2)$ is an open set as follows: Let $x \in (1,2)$ and $\delta = \min\{x-1,2-x\}$. Then ...
0
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1answer
52 views

Show that $\cap_{k\in \mathbb{N}}[-\frac{1}{k},k+1]$ is a closed set

I know that a subset $M$ of a metric space $(X,d)$ is open if it contains a ball about each of it points, and closed it its complement is open. But how would I show that the set $\cap_{k\in ...
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1answer
4k views

Continuous functions do not necessarily map closed sets to closed sets

I found this comment in my lecture notes, and it struck me because up until now I simply assumed that continuous functions map closed sets to closed sets. What are some insightful examples of ...
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1answer
295 views

Lipschitz continuity of an integral

Let $(E,d)$ be a metric space, $\mathscr E$ be its Borel $\sigma$-algebra and $\mu$ be a $\sigma$-finite measure on $(E,\mathscr E)$. Let the function $p:E\times E\to\mathbb R_+$ be non-negative and ...
2
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1answer
236 views

Completeness of $\ell^2$ space

I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a ...
4
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1answer
261 views

Quasi-Isometry [Geometric Group Theory]

How can I prove that if $S,S'$ are two different finite generating sets of a group $G$ , then the metric spaces induced by the "word metric" are quasi-isometric? The definition of quasi-isometry is: ...
0
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4answers
288 views

If $M$ is complete and $f : (M,d)\to(N,p)$ is continuous, then $f(M)$ is complete?

Prove or disprove: If $M$ is complete and $f:(M, d )\to (N, p)$ is continuous, then $f(M)$ is complete.
3
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2answers
422 views

Topological proof of Bolzano-Weierstrass

Below is my attempt to prove the topological version of the Bolzano-Weierstrass Theorem. Is it an effective proof? I'd appreciate any comments on it. The book gave a hint to use a nested sequence ...
4
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2answers
615 views

Definition of metrizable topological space

I am learning a bit about Topology through independent study. I am using Bert Mendelson's "Introduction to Topology - 3rd Edition". I have a question on one of the book's example and related ...