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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$M=X\cup Y$. 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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385 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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206 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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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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374 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 ...
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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) = ...
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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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912 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 ...
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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 ...
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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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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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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 ...
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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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880 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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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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389 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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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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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 ...
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53 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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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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306 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 ...
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261 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 ...
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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: ...
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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.
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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 ...
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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 ...
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Understanding the definition of continuity between metric spaces.

As we know that $\epsilon-\delta$ definition of continuity between metric spaces $X$ and $Y$ can be stated as follows: A map f:$(X, d_X)\rightarrow (Y, d_Y)$ is said to be continuous at a point ...
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1answer
201 views

Schauder's fixed point theorem for metric linear space

Is there an analogue of Schauder type fixed point theorems that can be used over a metric linear space. So, here $(X,d)$ is a complete vector space with metric $d$. If $C\subseteq X$ and ...
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Homeomorphism preserving distance

I have a problem but I don't know if there is a solution or a counter-example. Problem: Let $M$ be a non trivial compact connected metric space and let $f:M\to M$ be a homeomorphism. Show that there ...
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Is $d_1(x,y)=2|x-y|$ a metric space?

Im trying to check if $d_1(x,y)=2|x-y|$ and $d_2(x,y)=|x-y|^2$ are metric spaces. Im just not sure how to proceed with checking the triangle inequality property $d(x,y)\le d(x,z)+d(z,y)$. Is what I ...
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Are trigonometric polynomials dense in $C_{2\pi}^m$?

Let $m \in N$ be fixed and let $C_{2\pi}^m$ be a class of functions $f : R \rightarrow R$ of class $C^m$ and periodic with a period $2\pi$ with the following metric $$d(f,g)=\sum_{k=0}^m \sup_{\{x ...
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$f:[-r,r]\to M$ continuous iff $f\circ \pi$ continuous in $B[-r,r]$

Let $\pi: \mathbb{R}^2\to \mathbb{R}$ defined by $\pi(x,y)=x$. Let $M$ be a metric space. Prove that $f:[-r,r]\to M$ is continuous if and only if $f\circ \pi: B[0,r]\to M$ is continuous on the ...
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Metrization of topological space

Can you help me please with this question? Let $X$ be a non-empty set with the cofinite topology. Is $\left ( X,\tau_{\operatorname{cofinite}} \right ) $ a metrizable space? Thanks a lot!
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1answer
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Condition for an interval being contained in a subset of $\mathbb{R}$

I'd like some input on this problem. It's a different sort from what I've done before, and it's that sort of problem that (I think) feels so nicely intuitive that it's hard to decide if my proof is ...
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1answer
137 views

The interior of $\mathbb{R} \times \mathbb{Q}$

A question says, find the closure and interior of the sets $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R} \times \mathbb{Q}$. The answers say $\mathbb{R}^2$ and $\emptyset$ respectively for both. Why ...
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Equivalence of three properties of a metric space.

Another question about the convergence notes by Dr. Pete Clark: http://math.uga.edu/~pete/convergence.pdf (I'm almost at the filters chapter! Getting very excited now!) On page 15, Proposition 4.6 ...
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1answer
171 views

Coordinate change for metrics

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me. As far as my understanding goes, given a Riemannian metric ...
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125 views

Is $M=\{(x,y)\in (0,\infty )\times\mathbb{R} : y=\sin(\frac{1}{x}) \}$ a closed set in space $((0,\infty )\times\mathbb{R} ,\rho_{e})$?

Is $M=\left \{(x,y)\in (0,\infty )\times\mathbb{R} : y=\sin(\frac{1}{x}) \right \}$ a closed set in space $((0,\infty )\times\mathbb{R} ,\rho_{e})$, $\rho_{e}$ - Euclidean metric ? I think that open ...
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1answer
185 views

Question missing condition in Royden Exercise 7.42 b, about Baire Category

In Royden's Real Analysis P164 Q7.42b, It assumes that $X$ and $Y$ are complete metric spaces. Let $O$ be a dense open set in $X \times Y$. Assertion: Then there is a $G \subset X$ which is a ...
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2answers
529 views

Multiple choice question from general topology

Let $X =\mathbb{N}\times \mathbb{Q}$ with the subspace topology of $\mathbb{R}^2$ and $P = \{(n, \frac{1}{n}): n\in \mathbb{N}\}$ . Then in the space $X$ Pick out the true statements 1 $P$ is ...
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If $d$ is a metric and $f$ a function when is $d \circ f $ a metric? [duplicate]

Possible Duplicate: What operations is a metric closed under? Let $f: \mathbb R_{\geq 0} \to \mathbb R_{\geq 0}$ be a function and $d : X \times X \to \mathbb R_{\geq 0}$ a metric. I've ...
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Completeness of normed spaces

As earlier, I have received an answer from this site that Bolzano Weierstrass' theorem is true for finite dimensional normed spaces, but not for infinite dimensional spaces. This, in particular => all ...
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1answer
86 views

$A =\{ 1/(n+1): n \in \mathbb N \} $ is a nowhere dense subset

Prove that the set $A =\displaystyle \left \{ \frac{1}{n+1} : n \in \mathbb N \right \} $ is a nowhere dense subset of $\displaystyle{ \mathbb R }$. I have think two ways but I can't finish it. ...
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Dynamical system: hypothesis on metric functions

Let $E$ be a completely metrizable separable topological space and $\mathscr E$ be its Borel $\sigma$-algebra. Consider a measurable map $F:E\to E$ such that if $f:E\to \mathbb R$ is continuous and ...
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40 views

Steiner Tree Approximation

My question is about a subtlety regarding the $2$-approximation for the Metric Steiner Tree problem. The classical Metric Steiner tree problem: Given a metric space on $n$ points and a subset $S$ ...
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Open Balls in Metric Space.

I'm working with the metric space $(\mathbb{N}, \rho)$ where $\mathbb{N}$ is the set of natural numbers and $\rho(x,y) = |\frac{1}{x} - \frac{1}{y}|$. I'm considering the open balls on this metric. ...
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$\Bbb Q$ is not complete metrizable. [duplicate]

Possible Duplicate: Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space? How do you prove that the space of the rational numbers with the usual metric (from the ...
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Showing that $\cos(x)$ is a contraction mapping on $[0,\pi]$

How do I show that $\cos(x)$ is a contraction mapping on $[0,\pi]$? I would normally use the mean value theorem and find $\max|-\sin(x)|$ on $(0,\pi)$ but I dont think this will work here. So I think ...
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Show that the set of complex numbers is complete metric space

I know that the set of complex number is a normed linear with norm $\|z\|=|z|$. The induced metric is $d(z,w)=|z-w|$ for complex $z$ and $w$. But I want to prove that the set is complete.Thanks for ...
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$K$ compact and $\Omega$ is open, then $\inf\{\rho(x,x') \mid x \in K \textrm{ and } x' \in \Omega^c\} > 0$

I have to show the following: $(V,\rho)$ be a metric space, $K\subset V$ compact and $\Omega \subset V$ is open, then $d(K,\Omega^c) = \inf\{\rho(x,x') \mid x \in K \textrm{ and } x' \in \Omega^c\} ...