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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$\alpha$ exists so that for any points $x_n$ there is a point at average distance $\alpha$ from the $x_n$.

Let $X$ be a connected and compact metric space. Prove a real number $\alpha$ exists so that for every finite set of points $x_1,x_2,\dots, x_n\in X$ (not necessarily distinct) there exists $x\in X$ ...
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36 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open.

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
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+100

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
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30 views

Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
3
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225 views

Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. The following is my proof I'd like to know if it is correct. Proof: I will use the fact that ...
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1answer
23 views

Base of the Baire space [on hold]

Why base of the Baire space is countable? Because it is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points.
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17 views

Equivalence between properties of compactness for metric spaces

I am attempting here to show the equivalence between the following three statements for the metric space $(X,d),$ i) $(X,d)$ is compact, meaning every open cover admits a finite subcover ii) $(X,...
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3answers
42 views

does the condition “every open set is a countable union of closed sets” imply metrizability

In metric spaces, every open set is a countable union of closed sets. is the converse true? A topological space with the property "every open set is a countable union of closed sets" has to be ...
4
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1answer
79 views

Proof of the Arzelà–Ascoli Theorem

I'm stuck on a particular line of the proof of The Arzelà–Ascoli Theorem. In lectures, we have: $1.$ Defined equicontinuous as: Let $X$ be a metric space, $C(X) = \{f: X \rightarrow \mathbb{R}\...
4
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2answers
32 views

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 $...
2
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0answers
48 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 ...
0
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1answer
23 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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1answer
145 views

Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.

I really need help with this question: Prove that a metric space which contains a sequence with no convergent subsequence also contains an cover by open sets with no finite subcover.
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{|...
3
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1answer
46 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 \...
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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 ...
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2answers
32 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 $M\times K$...
0
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2answers
56 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: $\...
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1answer
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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 $...
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 ...
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1answer
49 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(...
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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 ...
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2answers
65 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 ...
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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 ...
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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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Definition of Cauchy Sequence

I have a question regarding the definition of a Cauchy sequence of a sequence in a metric space. The definition I learned and that is consistent with Wikipedia defines a sequence $(x_n)_{n=1}^\infty$ ...
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1answer
30 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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Compactness of the set of points where a continuous function achieves a local maximum

Let $(K,d)$ be a compact metric space, and $f:K\rightarrow \mathbb{R}$ be a continuous function on $K$. Define: $$M=\left \{ x\in K :\text{$f$ achieves a local maximum in $x$} \right \}$$ I need to ...
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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 ...
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1answer
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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 ...
2
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1answer
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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
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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
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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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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_\...
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4answers
79 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
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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
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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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1answer
604 views

Uniform convergence of a sequence of functions and equicontinuity

Let $(X,d)$ be a compact metric space. I would like to prove that if $(f_n)_{n \in \mathbb{N}}$ is a sequence of continuos functions $f_n:X \to Y$ that converge uniformly in $X$, then $(f_n)_{n \in \...
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 ...
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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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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: ...
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 ...
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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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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 $A_0,...,A_n \in \mathcal A$ ...
3
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1answer
760 views

Pearson correlation and metric properties

Assuming that the data set was $z$-standardized to zero mean and unit variance (also assuming that it does not contain constant vectors). Then Pearson's r reduces to Covariance: $$\rho(X,Y) := \frac{...
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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 $...
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5answers
700 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 $\mathbb R^2\setminus \{(0,0)\}$ ?
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3answers
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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?