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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4
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Identical Geodesics implies scalar multiple of metric?

Suppose $(M,g^1)$ and $(M,g^2)$ are two intrinsic metric spaces with the same underlying set $M$. Assume that for every $p,q\in M$, for each geodesic $\gamma^1_{[p,q]}$ connecting $p$ to $q$ under $...
0
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1answer
20 views

Prove that $(\mathcal{F}(E, \mathbb{R}), \|.\|_{\infty})$ is complete

Let $f_n$ be Cauchy for the $\|.\|_{\infty}$ norm, meaning we have $$\forall \varepsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, \forall p \in \mathbb{N}, \|f_n - f_{n+p}\|_{\infty} < \...
2
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0answers
38 views

The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
-2
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1answer
32 views

How do I show that $d:\mathbb{R^2}\times\mathbb{R^2}\rightarrow\mathbb{R}$ is a metric defined in $\mathbb{R^2}$? [on hold]

If $d(\vec{u},\vec{v}) = \lvert u_1-v_1\lvert+\lvert u_2-v_2\lvert$ for $\vec{u}=(u_1,u_2),\vec{v}=(v_1,v_2)$. How can I show that $d$ is defined in $\mathbb{R^2}$? Would it be enough to show the ...
1
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2answers
31 views

Question about proof of the tube lemma for metric spaces

Tube lemma: Let $M$ be a metric space and $K$ a compact metric space. Let $a\in M$, $a\times K\subset V\subset M\times K$, that is, suppose there is an open set $V$ between $a\times K$ and $M\times K$...
3
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1answer
43 views

Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
0
votes
1answer
16 views

Computing Hausdorff metric for some sets

Just started to learn about metric spaces, and I came across the Hausdorff metric. Let $K$ be the family of non-empty closed subsets of $[0,1]$. For $A \in K$ and $\delta > 0$ let $A_{\delta}$ be ...
0
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1answer
21 views

if the metric $d_1$ is complete, and $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$, is $d_2$ complete?

two metrics $d_1, d_2$ on $X$, For all $x_n$ and $x$ from $X$ it holds : $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$ Is it true that $(X, d_1)$ complete implies that $...
0
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0answers
14 views

Metric Matrix of the hyperbolic reimmanian manifold

Let $\Bbb{H}^n:=\{(x_1,...,x_n)\in\Bbb{R}^n|x_n>0\}$ be the hyperbolic space and $g={d^2x_1+...+d^2x_n \over x_n^2}$ be the standard hyperbolic metric. Looking at the $(\Bbb{H}^n,g)$ remannian ...
1
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1answer
47 views

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is a continuous function. I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(...
0
votes
2answers
54 views

$\phi:M\to \mathbb{R}$ continuous, $\phi(x)<\epsilon$ for $x\in X$, then $\phi(x)\le \epsilon$ for $x\in\overline{X}$

I was reading a proof that if a sequence of functions from $M$ to $N$, where $N$ is complete, converges uniformly in $X$, then they converge uniformly in $\overline{X}$, and it uses this result: $\...
2
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1answer
62 views

if $M$ is compact, then every continuous bijection $F:M\to N$ is an homeomorphism

My book proves that: if $M$ is compact, then every continuous bijection $f:M\to N$ is an homeomorphism by the following: Being $f$ closed, your inverse $g:N\to M$ is a function such that $F\subset ...
1
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0answers
29 views

Question about proof ot Tychonoff's theorem for metric spaces

Tychonoff's theorem: The cartesian product $M = \prod_{i=1}^{\infty}M_i$ is compact $\iff$ each $M_i$ is compact. My book, before proving it, says that the proof will happen like this: Given an ...
0
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2answers
64 views

$M\times N$ compact $\implies$ $M$ compact and $N$ compact

I must prove that $M\times N$ compact $\implies$ $M$ compact and $N$ compact using the definition that, if a metric space $M$ is compact, then every cover has an open finite sub cover. $$M=\cup ...
3
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2answers
33 views

Hints on showing Cauchy sequence converges

Let $T>0$ and $L\geq0$. Let $C[0,T]$ be the space of all continuous real valued functions on $[0,T]$ with the metric $\rho$ defined by $$\rho(x,y)=\sup_{0\leq t\leq T}e^{-Lt}\left|x(t)-y(t)\right|$...
1
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1answer
27 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
0
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2answers
25 views

Closure of sets (specifically regarding the notation)

I'm new to sets and the notation is somewhat confusing to me. I just want to see if what I'm doing makes sense. For the following sets I need determine if it is open, closed, or neither. I also ...
5
votes
1answer
51 views

For an arbitrary uncountable set of irrational numbers, can I always construct a sequence from them that converge in the rationals?

Suppose you have a set $S$ of uncountably many irrational numbers. Can you construct a sequence of $S$ that converges to a rational number? What I have tried: Since $S$ is uncountable, the inf of ...
2
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1answer
22 views

Defining compact sets with closed covers

This question is a continuation of this. My book says that a metric space is compact if and only if: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\...
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2answers
45 views

Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
0
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1answer
17 views

Hyperbolic distance of a point from center in Klein-Beltrami disk model

According to the Wikipedia entry about Klein Beltrami disk, I found that the hyperbolic distance between two points P and Q is determined by the following formula : $$d(P, Q) = \frac{1}{2} \ln \frac{|...
0
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1answer
27 views

compact metric space definition by closed covers

My book says the following: A metric space is compact iff: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\lambda}$ is open. Then, it says that if $A_\...
0
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1answer
41 views

CW complex= cell complex?; compact metrizable spaces

I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\...
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4answers
78 views

prove triangular inequality for $ d(x,y)= \frac{||x-y||}{1+||x-y||}$ [duplicate]

prove triangular inequality for $$ d(x,y)= \frac{||x-y||}{1+||x-y||}$$ that is $d(x,y) \leq d(x,z)+d(z,y)$ ofcourse ||.|| is a norm and has properties of norms this usually works $$ \begin{...
2
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1answer
51 views

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
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1answer
52 views

Metric in $\mathbb{S}^1$

Let $\mathbb{S}^1=\{ x=e^{2 \pi ir} | r \in I \}$, if $$d(x,y) = \left\{ \begin{array}{lcc} \min\{s-r,1-s+r\} & \text{if} & r \leq s \\ \\ \min\{r-s,1-r+s\} & ...
2
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0answers
33 views

Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
2
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2answers
69 views

Are these subsets homeomorphic?

Are the two subsets of the Euclidean Plane $[0,1]\times[0,1)$ and $[0,1)\times[0,1)$ homeomorphic or not? My attempt: We need to find a bijective function $f$ from $[0,1]$ to $[0,1)$ such that $f$ ...
0
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1answer
32 views

Portuguese term for “path metric”

Do anybody knows what is the usual translation to Portuguese for "path metric"? (Given a metric space $(M,d)$, $d$ is called a "path metric" if, given any pair $(x,y)\in M\times M$, there exists a ...
2
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2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
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2answers
25 views

Metric space where each continuous function has IVP is connected

The question: Let $X$ be a space such that every continuous function $f:X\rightarrow\mathbb{R} $ does have the following property: if $a<c<b$, $f(x) =a$, and $f(y) =b$, then there exists $z\in ...
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2answers
31 views

Union of path connected pairwise not disjoint subsets

Problem Let $(X,d)$ be a metric space and let $\mathcal A$ be a family of path connected subsets of $X$ such that for every pair of sets $A,B \in \mathcal A$ there are $A_0,...,A_n \in \mathcal A$ ...
0
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1answer
33 views

existence of certain function on unit interval

I'm trying to solve this exercise in an introductory book on general topology: Let $(X,d)$ be a metric space and $A,B \subset X$ disjoint closed subsets. Show that there exists a continuous function $...
0
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3answers
45 views

Is there a complete metric space which has no Cauchy sequence?

Definition: A metric space is said to be complete if every Cauchy sequence is convergent. Now, my question is: Is there a complete metric space which has no Cauchy sequence?
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2answers
45 views

problem in real analysis about open sets in metric spaces.

For $x = (x_1, x_2, \ldots, x_n)$ and $y = (y_1, y_2, \ldots, y_n)$ in $\mathbb{R}^n$. Let $d_p(x, y) = \Bigg(\sum\limits_{i=1}^n |x_i-y_i|^p\Bigg)^\frac{1}{p}$ for $1 \leq p < \infty$ and $d_\...
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1answer
24 views

Cauchy sequence in $\mathbb{R}^d$

Is is possible to prove that sequence $\{x_n\}$ with terms in $\mathbb{R}^d$ has limit iff $\forall_{\epsilon > 0}\exists_{N \in \mathbb{N}}\forall_{n,m \ge N} \rho(x_n, x_m) < \epsilon$ as a ...
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1answer
59 views

Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ? [closed]

Is $\mathbb R^2 \setminus D^2$ , where $D^2=B[0;1]$ is the closed unit disk , homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ?
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5answers
697 views

Is the plane minus a line segment homeomorphic with punctured plane?

Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\}) $ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?
0
votes
1answer
41 views

Show that $E \subset \Bbb Q$ is closed in $(\Bbb Q, d)$

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} \subset \Bbb Q.$ I have to show that $E$ is closed. I see two ways of proving ...
2
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1answer
34 views

Can an arbitrary metric space be made into a complete and separable metric space?

Can any metric space be made into a separable and complete metric space by suitably choosing an isometry?
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1answer
40 views

On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
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2answers
77 views

$f$ be a function on real line carrying compact sets to compact sets and fiber of every point under $f$ is closed , is $f$ continuous ?

Let $f:\mathbb R \to \mathbb R$ be a function such that it carries compact sets to compact sets and $f^{-1}(\{x\})$ is closed for every $x \in \mathbb R$ , then is $f$ continuous ? (I know that if $...
13
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1answer
168 views

If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded ...
0
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1answer
65 views

$T:\mathbb R^n \to \mathbb R^n $ be an isometry , is $T$ surjective?

Let $T:\mathbb R^n \to \mathbb R^n $ be an isometry and $T(0)=0$ , then $T$ is linear and $T(B[0,1])\subseteq B[0,1]$ so $T:B[0,1]\to B[0,1]$ is an isometry and since $B[0,1]$ is compact so $T|_{B[0,...
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0answers
16 views

For what functions $f$ is $d_f$ defined by $d_f(x,y) = f(d(x,y))$ also a metric on $X$ assuming $d$ is a metric on $X$

For what functions $f : \mathbb{R}_{\leq 0} \to \mathbb{R}_{\leq 0}$ is it true that for every metric $d$ on a set $X$, the function $d_f$ defined by $d_f(x,y) = f(d(x,y))$ is also a metric on $X$? I ...
1
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1answer
43 views

$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
2
votes
1answer
35 views

Verification of proof, interior is open.

Let $(\mathbb{X},d_{\mathbb{X}})$ be a metric space and $\emptyset \neq A \subseteq \mathbb{X}$ its subset. Prove that the interior $A^{\circ} =\{ a \in A | \exists\epsilon(a) > 0, B_{\epsilon}^{d_{...
0
votes
1answer
36 views

In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance
2
votes
1answer
84 views

Proving that the mapping $ (Ff)(t): C[0, 1] \to C[0, 1] $ is a contraction

This is a follow-up to my previous question. I tried to use John Ma's answer, but couldn't solve this. I need to prove that $F : C[0,1]\to C[0,1]$ is a contraction mapping. $ F $ is defined as ...
6
votes
2answers
105 views

$f$ non-constant on $\mathbb R$ such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , is $f$ identity?

Let $f:\mathbb R \to \mathbb R$ be a non-constant function such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , then is $f$ the identity function i.e. $f(x)=...