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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$M\times N$ 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 ...
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2answers
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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|$...
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32 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\...
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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 ...
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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 ...
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1answer
24 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
51 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 ...
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1answer
20 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{|...
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1answer
28 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_\...
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1answer
45 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
80 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{...
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1answer
53 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\} & ...
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0answers
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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: ...
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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 $f$ ...
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1answer
34 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 ...
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2answers
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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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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 $A_0,...,A_n \in \mathcal A$ ...
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1answer
35 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 $...
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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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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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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 $\mathbb R^2\setminus \{(0,0)\}$ ?
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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 ...
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35 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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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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$f$ 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 $...
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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 ...
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1answer
66 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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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 ...
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1answer
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$B$ 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 ...
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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_{...
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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
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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 ...
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$f$ 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)=...
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To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$...
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1answer
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Product spaces from metric induced topological spaces $(X,T) \times (X,T)$ into $\Bbb R$ is continuous

Let $(X,d)$ be a metric space and $T$ the induced topology on $X$ by $d$. Prove that the function $d$ from the product space $(X,T) \times (X,T)$ into $\Bbb R$ is continuous. My question is what ...
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$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ?

Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=...
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Existence of metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is everywhere discontinuous

Does there exist a metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is everywhere discontinuous ? It is motivated from this question which ...
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0answers
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A set is compact iff every collection… Proof check

I asked this question (A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection) a few days ago and was lucky enough to get an answer, but I'm afraid that the ...
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1answer
36 views

Infinite Union of Complete Metric Subspaces which is Not Complete

Can anyone think of an example of a metric space $(X,d)$ and an infinite set of complete metric subspaces in $(X,d)$ such that their union is not complete?
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28 views

Example Infinite Product of Compact Metric Spaces that is not itself compact [closed]

Can anyone think of an infinite product space of compact metric spaces that us not itself compact?
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2answers
33 views

Showing this a norm

I want to show that $$\| x \| = \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{\left| x_n \right|}{1+\left| x_n \right|}$$ is a norm. I'm fine showing positivity and the triangle inequality, to show the ...
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1answer
34 views

$\ell_0$ norm and the induced complete metric spaces

I have been reading about the $\ell_0$ norm, wikipedia gives us that "The mathematical definition of the $\ell_0$ norm was established by Banach's Theory of Linear Operations. The space of sequences ...
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0answers
14 views

The meaning of “order of congruence” of metric space

I was studying low-distortion embedding of finite metric space, and was confused about the following concept: Order of congruence: A metric space $(X,D)$ has order of congruence at most $m$ if every ...
3
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2answers
83 views

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is not continuous?

Does there exist a metric $d$ on $\mathbb R$ such that the map $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is not continuous?
2
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1answer
60 views

subset of a topological space is closed if and only if it contains all of its limit points.

I'm trying to prove the following: Show that a subset of a topological space is closed if and only if it contains all of its limit points. Is my proof valid? Definition of limit point: $p$...
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1answer
31 views

If union and intersection of two subsets are connected, the subsets are connected

I've been able to prove what is proved here If union and intersection of two subsets are connected, are the subsets connected? However, I was wondering if I could get some help finding an example to ...