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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773 views

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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2answers
238 views

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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99 views

$f:[-r,r]\to M$ 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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2answers
293 views

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
52 views

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 ...
2
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1answer
131 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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3answers
400 views

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
154 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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121 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
147 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
304 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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0answers
129 views

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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2answers
265 views

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
78 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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1answer
59 views

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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39 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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2answers
661 views

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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0answers
66 views

$\Bbb Q$ 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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1answer
848 views

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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3answers
768 views

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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1answer
100 views

$K$ 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\} ...
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175 views

A metric space is path connected and countable then it is complete

I have to show that if a metric space is path connected and countable then it is complete. I'm pretty lost where to start this at all. I have the basic definitions of complete, path-connected, compact ...
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49 views

existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following: Is there exists space $Z\subset Y$, such that ...
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1answer
336 views

Metric on $\Bbb{R}^n$ which comes from a continuous function

I've been struggling to prove the following fact for some time now, and I didn't manage to do so. Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros ...
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2answers
265 views

Contraction Mapping question

Let X be the set of continuous real valued functions defined on $[0,\frac{1}{2}]$ with the metric $d(f,g):=\sup_{x\in[0,\frac{1}{2}]} |f(x)-g(x)|$. Define the map $\theta:X\rightarrow X$ such that ...
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1answer
133 views

Dynamics Question

Let be $T_{\beta}:[0,1]\to [0,1]$ defined by $T_{\beta}(x)=\beta x \bmod 1$ where $\beta \in (1,2).$ Questions: $T_{\beta}$ is topologically transitive? What about the periodic points? ...
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3answers
132 views

A characterization of open sets

Let $(M,d)$ be a metric space. Then a set $A\subset M$ is open if, and only if, $A \cap \overline X \subset \overline {A \cap X}$ for every $X\subset M$. This is a problem from metric spaces, but ...
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255 views

product of hermitian and unitary matrix

Could anyone tell me how to show that, for any $g\in GL_n(\mathbb{C})$, $\exists$ $R$ a hermitian matrix with positive eigenvalues and $U$ an unitary matrix such that $g=RU$? And (I am not sure) can ...
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244 views

Open Dense Subset of $M_n(\mathbb{R})$

Well, I know the fact that $GL_n(\mathbb{R})$ is open set in $M_n(\mathbb{R})$, how to show that it is dense also? Well I thought like this: If $A\in M_n(\mathbb{R})$ and If ...
3
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0answers
127 views

Sequences of Metric Spaces of Compact Subsets

Consider a complete metric space $(M, d)$ and let $F(M)$ denote the non-empty compact subsets of $M$. Then $F(M)$ is also a complete metric space under the Hausdorff distance $d_H$. Given some ...
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1answer
386 views

Proving that if $u \in A$ is an upper bound of $A$, then $u = \sup A$

Let $A\subset\mathbb{R}$ a nonempty set of real numbers bounded above and $u$ be an upper bound of $A$. Prove that if $u\in A$, then $u=\sup A$.
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1answer
577 views

Definition of Basis for the Neighborhood System

I'm trying to learn a bit about topology through independent study. I've been using Bert Mendelson's "Introduction to Topology - 3rd edition". I'm having a lot of fun but I'm a bit confused regarding ...
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2answers
300 views

Pete L. Clark's Convergence Notes

I had initially sought out a better understanding of filters and nets, and a few quick google searches showed this document as highly recommended. (And they are excellent!) I'm having a bit of ...
2
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1answer
65 views

$\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold

Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth. I ...
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1answer
218 views

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
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1answer
54 views

Product of Transitive Systems

Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$. We say that a dynamical system, $f:M\to M$ is topologically transitive when, ...
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185 views

Is a metric on a metric space a bilinear form?

I've just finished a course on bilinear forms and am now starting a cause on topological spaces and was just wondering; for a metric space which is made up of a set $M$ and a metric function $d$ such ...
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262 views

$\epsilon$- dense subsets

Let be $M$ a compact metric space, and let $\{x_n\}$ be a dense subsequence in $M$. We say that a set $\Lambda=\{y_1,\ldots,y_n\}$ is $\epsilon$-dense when every ball of radius $\epsilon$ ...
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1answer
82 views

the sphere $S^n$ is a metrically homogeneous

A metrically homogeneous space is a metric space $(X,d)$ such that for all points $p$ and $q$ in $X$, there exists an isometry $f$ such that $f(p) = q$. Does the sphere $S^n$ have this property? odd ...
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236 views

If two metrics have the same Cauchy sequences, does that imply uniform equivalence?

If two metrics $d_i$ on the same set $X$ have the same Cauchy sequences (ie. if a sequence is Cauchy for the first metric, it is also Cauchy for the other one and vice versa), can we conclude that the ...
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1answer
298 views

closed unit ball with radius 1

Is this subset compact in $l_1$ of all absolutely convergent real sequences, with the metric:$d_1(\{a_n\},\{b_n\})=\sum_{1}^{\infty}|a_n-b_n|$ closed unit ball centered at $0$ with radius $1?$ I ...
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1answer
204 views

P-adically Cauchy sequences

I am trying to do the following question Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$. In general in $\mathbb{Q}_p$ what is the stronger condition, to be ...
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1answer
464 views

A closed subset of continuous functions on [0, 1]

How would one show that the set consisting of the monomials $1,x,x^2,...$ is a closed subset of the metric space $C[a,b]$ under the metric $d(a,b) = ||a-b|| =sup_{[0,1]}|a-b|$ ? I considered its ...
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2answers
434 views

Analogue to Fixed Point Theorem for Compact metric spaces

If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
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1answer
79 views

Show that $D: C^1([a, b]) \mapsto C^0([a, b]): f \mapsto f'$ is continuous.

the problem I have to show that a function $D: C^1([a, b]) \mapsto C^0([a, b]): f \mapsto f'$ is continuous given a metric $\| \cdot \|_{C^1([a, b])}$. The metric $\| \cdot \|_{C^0([a, b])}$ is ...
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1answer
126 views

Equilibrium distance formula proof

Let $$d: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}$$ be defined by $$d(x_i,x_j)=\frac{|x_i-x_j|}{\sqrt{M(i)M(j)}},$$ where $M(i)$ represents the average distance between $x_i$ and ...
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1answer
80 views

Coordinates translation in space

First of all sorry if the title is somewhat opaque, the problem I am trying to solve is already hard to explain properly in my first language. So, let's consider we have a plane, rectangle target in ...
3
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1answer
312 views

Relationship between Minkowski distance and Minkowski space

The metric induced by the p-norm: $d((x_1,\dotsc,x_n),(y_1,\dotsc,y_n)) = \left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}$ is often called the Minkowski distance. There is also Minkowski space, which ...
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1answer
80 views

Every cover covers sets with $\mathrm{diam}(A) < \lambda$ [duplicate]

Possible Duplicate: Proof of the Lebesgue number lemma Let $(X, d)$ be a metric space and $K \subset X$ a compact set. Now I have to show that for all open covers $\mathcal U$, there is an ...
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2answers
561 views

The union of open balls.

Question Show that every open subset of a metric space can be expressed as a union of open balls. So far I have the following: "Let $U \subseteq X$. For each $a \in U$, choose $r_a > ...