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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Metric invariant under translations is projective

Show that a metric d on the plane that is invariant under translations is automatically projective. Note that we only consider length spaces, which are metric spaces where the distance between two ...
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Ostrowski: an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$. I will ...
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1answer
29 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
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2answers
34 views

Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$

Let $(X,d)$ be a metric space. Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$. If $F\subseteq X$ is closed then $K\subseteq X$ compact implies $K$ ...
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5 views

p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
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1answer
59 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
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2answers
34 views

On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...
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1answer
19 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
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1answer
26 views

If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
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131 views

$\epsilon$-dense subsets on $\mathbb R/\mathbb Z$.

Let $\langle M, d\rangle$ be a metric space. We say that $A \subset M$ is $\epsilon$-dense if every open ball of radius $\epsilon$ contains a point of $A$. Now let $T=\mathbb R/\mathbb Z$, the ...
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1answer
20 views

Sufficient conditions for embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.

This is a followup to a question I asked in this thread. I'm posting separately so points can be awarded. Hopefully someone can help me with a reference for this problem, or the construction. I ...
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44 views

On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive-definite $n\times n$ real matrices. I am looking for hints concerning the visualization of such spaces for $n=1,2,\ldots$. I know that ...
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1answer
27 views

Two exercises about hypermetric spaces

Take $S$ to be the collection of all subsets of $\{1,\dots,n\}$. If $x, y$ are in $S$, define $d(x,y)$ as the number of elements of the symmetric difference $x\triangle y$. Exercise 2.1. Show ...
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1answer
27 views

Embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.

Hopefully someone can help me with a reference for this problem, or the construction. I have a metric defined on $n$ points in $\mathbb{R}^2$. Is it possible to find a higher dimensional Euclidean ...
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18 views

Any compact metric space is Borel equivalent to some subset of $[0, 1]$

In Petersen's Riemannian Geometry book I encountered the following statement : Any compact metric space $X$ is Borel equivalent to some $S \subset [0, 1]$ i.e. there is a bijection $f : X \rightarrow ...
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1answer
47 views

Prove that the convergent sum of a real sequence is a metric

I want to show that $$ \varrho(\{a_n\},\{b_n\})=\left(\sum_{n=0}^\infty{(a_n-b_n)^2}\right)^{1/2} $$ is a metric, where $\{a_n\}_{n\in\Bbb N}\in \ell_2$, and $\ell_2$ is the set of all real sequences ...
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1answer
16 views

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?
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1answer
54 views

In $\mathbb Q_p$, proving every open ball is the disjoint union of more than one open ball

I'm reading the Foundations chapter of Gouvea's p-adic Numbers: An Introduction, and I'm trying to solve the following problem he poses to the reader: Take the $p-$adic absolute value on $\mathbb ...
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4answers
29 views

Show that if $X$ is sequentially compact, then $X$ is complete and totally bounded

Given a metric space $X$ which is sequentially compact (i.e every sequence has a converging subsequence), show that $X$ is complete and totally bounded. I've already shown that $X$ is complete, since ...
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38 views

Homeomorphism from unit ball to unit sphere [on hold]

Consider the unit sphere in $\Bbb R^3$ given by $\{(x,y,z) \in \Bbb R^3|x^2+y^2+z^2 =1\}$. Let $p$ be a point in this unit sphere. Question: How can I construct an open set U within this unit sphere ...
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1answer
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$A$ be a non-empty closed convex subset of a Hilbert space $H$ , is the distance from $A$ always attained at a unique point in $A$ ?

Let $A$ be a non-empty closed convex subset of a Hilbert space $H$ , then is it true that for every $b \in H$ , $\exists$ unique $x_b \in A$ such that $||x_b-b||=d(b,A)=\inf \{||b-x||:x \in A\}$ ?
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1answer
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A question on metric spaces which does not have Lebesgue covering property

Let $X$ be a metric space , $\{U_{\alpha}\}$ be an open cover of $X$ which has no Lebesgue number . So for every $r>0 $ , there is an open ball of radius $r$ which is not contained in any open set ...
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1answer
30 views

Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
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1answer
28 views

Prove that $G(f)$ is homeomorphic to $X$.

Let $X,d$ be a metric space .Let $f:X\to \mathbb R$ be a continuous function.Define $G(f)=\{(x,f(x)):x\in X\}$. Prove that $G(f)$ is homeomorphic to $X$. My try: Since $f$ is continuous then $G(f)$ ...
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2answers
64 views

Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow ...
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21 views

Metric spaces and compactness [on hold]

Let $X$ be a metric space. If for all compact $K$, the set $K\cap F $ is closed, then $F$ is closed.
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4answers
286 views

Little confusion about connectedness

Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$. So, $\overline X$ is also connected , as closure of connected set ...
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31 views

Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
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60 views

Is $[-1,1]$ complete under the Euclidean metric? [closed]

Is it true that the interval $[-1,1]$ is complete under the Euclidean metric?
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32 views

Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$

Let $(x_n)$ be a sequence in $X$ and $x \in X$. $(X,d)$ is a metric space and $f: X \times X\to $ is a bijection. Show that $x_n \to x$ in $(X,d_f)$ iff $f(x_n) \to f(x)$. ...
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2answers
31 views

Show that $d_f$ is a metric on $X$ [closed]

Let $(X,d)$ be a metric space, and let $f: X \to X$ be a bijection. Define $$d_f: X \times X \to \mathbb R $$ as $d_f(x,y)=d(f(x),f(y))$ $\forall x,y \in X$ Show that $d_f$ is ...
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1answer
40 views

What exactly is the distance of two elements in $C[0,1]$?

If $C[0,1]$ — the set of all continuous functions from $[0,1] \rightarrow \mathbb R$ — is equipped with the metric $||\cdot||_1$ (1-Norm), then what is the distance between ...
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2answers
54 views

Bounded complete metric space is compact?

This question may seem trivial, but in topology we were taught that in a complete metric space, a subset of that space was compact if and only if it is closed and bounded. Moreover, we are told that ...
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41 views

Is $ \text{Int} \overline{B(a;r)} = B(a;r)$ for a metric space $(X,d)$?

I think this is true in general. To give a brief outline of a proof: Let $ \text{Int} \overline{B(a;r)} = U $, I claim that if $a \in U \implies a \notin Fr(B(a;r))$ so $a \in \text{Int}B(a;r)$ ...
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24 views

Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
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2answers
112 views

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
3
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2answers
61 views

If $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected.

I'm trying to show that if $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected. First of all, I think I have to assume that $A$ and $B$ are nonempty, or else the statement ...
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58 views

A subspace of a metric space is normal

Is it true that every subset $Y$ of a metric space $X$ is a normal topological space? I think the answer is yes, because $Y$ is a metric subspace of $X$ equipped with the induced metric by the one of ...
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1answer
23 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
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1answer
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Normed space of bounded functions $f:\mathbb{N}\to\mathbb{N}$

Let $X = \{f:\mathbb{N}\to\mathbb{N}: \exists M\in\mathbb{N} \forall n\in\mathbb{N} f(n) \leq M\}$. Define a norm on $X$ by defining for $f\in X$: $$||f|| = \sum_{n=1}^\infty \frac{f(n)}{2^n}.$$ Is ...
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38 views

Prove that a finite union of closed sets is also closed (using limit points)

Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed. Proving that statement is equivalent to proving: If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then ...
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1answer
33 views

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact?

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again ...
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0answers
31 views

Lower semicontinuity on a metric space

I'm trying to prove something about lower semicontinuity for a map on a metric space $(X,d)$. I will try to write here my idea of the proof, hope someone can approve or contest it. Def. Let $(X,d)$ ...
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2answers
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Dense-in-itself open sets in a subspace of the real line

Given an uncountable set $X\subset [0,1]$ it is easy to write $X$ as a disjoint union of a perfect set $P$ (perfect in the subspace $X$) and an at most countable set $C$: just take $P$ as the set of ...
2
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2answers
59 views

If $f:X \to [0,1]$ be an onto continuous map and $\{f^{-1} (y)\}$ is compact then Is $X$ compact?

If $f:X \to [0,1]$ is an onto continuous map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again $X$ is ...
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34 views

Topologically equivalent metrics, using different definitions.

I´ve been dealing with topologically equivalent metrics for a while, using the usual definition, that $d$ and $d'$ are topologically equivalent iff they have the same open sets. However, there is ...
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1answer
35 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...
3
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1answer
35 views

Show that $A=\bigcap G_{A}$

Given a metric space $(X,d)$ and $A\subset X$, let $G_{A}$ be the set which consists of all the open sets that contain $A$. Show that $A=\bigcap_{B \in G_{A}}B$ It is obvious that $A \subset ...
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2answers
36 views

Is $d_1(x,y):= x^2-y^2$ a metrics on $\mathbb{R}$? [closed]

$$d_1(x,y) = x^2-y^2 \quad \forall x,y \in \mathbb{R}$$ Is $d_1$ a metric on $\mathbb{R}$?
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1answer
21 views

show that $d(a,b)\leq r-s-t \Rightarrow K(c,t) \subseteq K(a,r)$ in a metric space, assuming that $r,s,t>0$ and $c \in K(b,s)$.

Alright, so in a metric space, $M$, with $r,s,t>0$, $a,b,c \in M$ and $c \in K(b,s)$ I have to show, that: $d(a,b)\leq r-s-t \Rightarrow K(c,t) \subseteq K(a,r)$ I really have no idea where to ...