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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Is a uniquely geodesic space contractible? II

Is a uniquely geodesic space, contractible ? With the extra assumption that closed metric balls are compact, there is an answer here. We expect here an answer beyond this extra assumption ...
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How is $xy=1$ closed in $\Bbb{R}^2$?

I read somewhere that $xy=1$ is a closed set in $\Bbb{R}^2$. A closed set is defined as the complement of an open set, or one which contains all its limit points. In metric spaces, it is defined as ...
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118 views

Triangle inequality of a nasty metric $\rho(A,B)=\max\left\{\sup_{x\in A}\inf_{y\in B}|x-y|,\sup_{y\in B}\inf_{x\in A}|x-y|\right\}$

This was a qual problem in Winter 2012 at my university. I feel comfortable verifying everything but the triangle inequality. Let $\Omega$ denote the set of all nonempty closed subsets of $[0,1]$ ...
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1answer
121 views

Is a uniquely geodesic space contractible? I

Is a uniquely geodesic space contractible ? We assume in addition that closed metric balls are compact. A post without this extra assumption is here.
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Is a closed and bounded subset of any complete metric space compact?

We know that every closed and bounded subset of $\Bbb{R}$ is compact. The proof proceeds by bifurcating $[a,b]$, and then using the property that in a complete metric space the infinite intersection ...
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424 views

Minimum distance between compact sets in complete metric space

Let $X,Y$ be compact sets in a complete metric space. Prove that there exist $x\in X,y\in Y$ such that $d(x,y)$ is a minimum. For any $x_0\in X$, consider the function $f(y)=d(x_0,y)$ for $y\in ...
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78 views

A basic question on limit point metric space etc

Suppose we have a metric space $X$ and we have an order on $X$. Now, let $E$ be a subset of $X$ which is bounded above. Now let $E'$ be the set of all limit points of $E$. If $E' \neq \emptyset$ then ...
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2answers
222 views

closed subset of $\Bbb R$ shows least upper bound property

Can we prove without using the least upper bound property in $\Bbb R$ that any closed subset $E$ of $R$ which is bounded above has a least upper bound in $\Bbb R$. If a subset of $\Bbb R$ contains all ...
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Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
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1answer
169 views

Homotopy problem for infinite dimensional topological space III

This post here is a specification of this post. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : $X_{n}$ is a $n$-dimensional regular CW complex. ...
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94 views

Is closed subset of $\Bbb R$ has least upper bound in $\Bbb R$

Closed subset of $\Bbb R$ has least upper bound in $\Bbb R$. Is this statement correct ? What about the set of all integers ? It is closed in $R$ but I don't see any least upper bound (or any upper ...
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216 views

How to prove that $\|x\|_q\le \|x\|_p$ if $p \le q$

I was reading this: Proving an inequality with $\|x\|_p$ metrics? Since it is a really old post, and I'm not sure that I'll get an answer, I hope asking this way, as an independent post, doesn't ...
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3answers
237 views

Uniquely geodesic spaces

The purpose of this list issue is to better understand the class of uniquely geodesic spaces. I'm looking for two different things : Overclass : for example geodesic space or contractible space. ...
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3answers
509 views

Find the farthest points in d-dimensional space

We have $n$ points with $d$ coordinates each and we want to find two of them for which distance between them is the biggest, in Manhattan metric. The obvious algorithm has complexity $O(n^2 \cdot d)$ ...
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2answers
80 views

A basic question on open cover and compact sets

Let $u=\{u_\alpha\}$ be an open cover of $[a,b], a <b $. Let $S=\{r \in [a,b]$ such that $[a,r]$ is covered by some finite collection of open sets belonging to $u\}$. Is $S$ non-empty ? How ?
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3answers
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If a connected set $A$ is contained in a set $B$ that is contained in $\mathrm{cl}(A)$, then $B$ must be connected

I don't follow the argument in line 3-4 that reads "Since $U$ is an open set containing $x$, there would be a point $y$ in $U \cap A$." If this line is indeed logical, could someone please elaborate ...
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949 views

If $(M,d)$ is a compact metric space and $f:M\rightarrow M$ is an isometry ($d(x,y)=d(f(x),f(y))$ for any $x,y\in M$), then $f$ is a homeomorphism [duplicate]

If (M,d) is a compact metric space and f:M→M is an isometry (d(x,y)=d(f(x),f(y)) for any x,y∈M ), then f is a homeomorphism. I proved that $f$ is continuous as its inverse,I proved that it is ...
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64 views

A basic question on limit point neighbouhood interior point etc

Can a limit point $p$ of a set $E$ $(p \notin E)$ be an interior point of $closure(E)$ ? Now, if $p$ has a neighbourhood contaning only $p$ then in that case it becomes an interior point of $E$. But, ...
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1answer
356 views

Hausdorff dimension of the set of rational numbers within a certain interval?

Intro: The Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. In general the Hausdorff dimension ...
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27 views

Elements of $D_{2n}$ in terms of isometries

In course of studying Dihedral Group I'm having trouble to get what exactly the elements of $D_{2n}$ are. According to the Dummit-Foote texts For each $n∈\mathbb Z^+,n≥3$ let $D_{2n}$ be the set ...
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1answer
186 views

Set of infinite tuples is closed, bounded, non-compact

Let $\mathbb{R}^{\infty}$ be the set of all "infinite-tuples" $x=(x_1,x_2,\ldots)$ of real numbers that end in an infinite string of $0$'s. Define an inner product on $\mathbb{R}^{\infty}$ by the ...
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1answer
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Proof that a discrete metric is indeed a metric space

QUESTION Let X be any set and $d : X \times X \to \mathbf{R}$ be given by $$ d(x,y) = \begin{cases} 0, & \text{if $x = y$} \\ 1, & \text{if $x \neq y$} \\ \end{cases}$$ Show that $d$ is a ...
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How to determine what kind of Cauchy sequences lie in a given space?

I understand the main principles of Cauchy sequences and metric spaces, but I have a particular question about determining whether or not a space is a complete space. If a space has all cauchy ...
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1answer
66 views

A basic question on closure of a set in metric space

Let $A_1, A_2, A_3,\dots$ be subsets of a metric space. I see in some excercise that closure of infinite union of $A_i$s is a superset (proper) of the infinite union of each of $A_i$s closure whereas ...
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3answers
197 views

Example of a set for which none of the limit points are in the set?

Is there any example of a set for which none of the limit points are in the set ? I can't think of such set right now.
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81 views

Proof on if a set is discrete

I would like to know how well I answered the following proof: was it concise? Was it elaborate/rigorous? Did I use incorrect notation? I would also like to know if the set is a $T_{1}$ space, such ...
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2answers
80 views

Proof of completeness

I have to prove that $(C^1[0,1],d_1)$ is complete metric space, where $d_1(f,g)=\max|f(x)-g(x)|+\max|f'(x)-g'(x)|,x\in[0,1]$ Firstly, I take an arbitrary Cauchy sequence of functions from ...
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2answers
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Is this proof that all metric spaces are Hausdorff spaces correct?

Let $x$ and $y$ be distinct points of a metric space $M$. Prove that there exist in $M$ disjoint open sets $U$ and $V$ with $x \in U$ and $y \in V$. Let $U$ and $V$ be open balls centered at $a$ and ...
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331 views

Metric spaces and openness

I am asked to prove two things. I would like to know if the proof was elaborate and concise. I would also like to know if proving reductio ad absurdum is looked down upon. I have heard from my ...
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106 views

If $A$ is everywhere dense in the metric space $M$, then the only closed set which contains $A$ is $M$.

If $A$ is everywhere dense in the metric space $M$, then the only closed set which contains $A$ is $M$. My attempt: Let $B$ be a closed set which contains $A$. My aim: Prove $B=M$ Since $A$ is ...
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143 views

Define a metric based on a topology

Is their a systematic way, based on a topology meterizable $(X, t)$ to define or compute some metric $d$ on $X$ such that the open balls in $(X, d)$ is a metric of $(X, t)$?
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42 views

How to decide if these two maps are proper?

We define a mapping $f$ of a topological space $X$ into a topological space $Y$ to be proper if the subspace $f^{-1}(C)$ is compact in $X$ whenever $C$ is a compact subspace of $Y$. Now how to ...
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Metric Space (Elementary Analysis) [closed]

Does there exist any continuous bijection from $\mathbb Q$ to $\mathbb Q\times\mathbb Q$? Explain why.
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1answer
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$A$ is a set of points in ametric space $M$ and $B$ is the set of all accumulations points of $A$. Prove that $B$ is closed.

$A$ is a set of points in a metric space $M$ and $B$ is the set of all accumulations points of $A$. Prove that $B$ is closed. Aim: Prove that $B^{\complement}$ is open. Let $y \in B^{\complement}$. ...
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1answer
70 views

Proof of the Banach Fixed Point Theorem

I am presenting to you my book's verion of the proof of the Fixed Point Theorem: $\{T^i x_0\}$ has been shown to be a cauchy sequence. As we have a complete metric space, this cauchy sequence has a ...
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1answer
106 views

A theorem on restriction of a metric

Consider $(X,d)$ be a metric space. Let $Y\subseteq X$ be a metric on itself and $E\subseteq Y$. Then there is a theorem which says that $E$ is open in $Y$ iff $E=Y\cap G$ for some open subset $G$ of ...
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Can a neighbourhood of a point be an singleton set?

Can a neighbourhood of a point be a singleton set containing that point only ? I think yes.
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36 views

Ultra-metricity in sets

Suppose there is a set $A$ equipped with a trivial distance function. Take $D(a, a) = 0, D(a, b) = 1, a \not= b \in A$. A set $A$ is ultra-metric if $D(a, c) \le \max[D(a, b), D(b, c)]$ for all $a, ...
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Using axioms to define metric spaces

Let $M$ be a set with three elements: $a$, $b$, and $c$. Define $D\colon M\times M\to[0,\infty)$ so that $D(x, x) = 0$ for all $x$, $D(x, y) = D(y, x)$ for $x \ne y$. Say $D(a, b) = r$, $D(a, c) = s$, ...
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1answer
91 views

Proving a set is a metric space

I am currently doing questions from Kaplansky's Set Theory And Metric Spaces. I come to seek validation on my answers because my book does not have an answer key. I am looking on ways to strengthen my ...
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118 views

Is this proof of uniqueness of the limit correct?

I've tried to show the following: let $(M,d_M)$ and $(N,d_N)$ be metric spaces and $f : M \to N$. If $a \in M$ and $\lim_{p\to a}f(p)$ exists, then it is unique. I'm a little unsure if the proof is ...
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1answer
214 views

Is it possible to do calculus on any field with a topology?

I'll try to make my point clear: when we consider the field of complex numbers $\mathbb{C}$ we can do calculus there because we have properties of a field and in the same time we have a topology to ...
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2answers
119 views

A basic question based on the definition of limit point, closed set etc.

Let $(X,d)$ be a metric space and $E$ be a subset of it. Now, for any neighbourhood of $p$ there exists a point $q, q \neq p$ and $q$ is a limit point of $E$. Then I have to prove that $p \in E$. How ...
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58 views

Metric Space (Elementary Analysis) [duplicate]

Let $X \subseteq \mathbb{R}^{n}$ be given by $$ X = \left\{ (b_{1}, \ldots, b_{n}) \in \mathbb{R}^{n} \mid \sum\limits_{i=1}^{n} \frac{b_{i}}{i} = 0 \right\}$$ Then prove that $X$ is closed in ...
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84 views

Proving a particular subset of $R^n$ is closed

Let $S,X$ be subsets of $R^n$ given by $$S=\{(a_1,a_2,\dotsc,a_n)\in R^n|\sum a_i^2=1\}$$ $$X=\{(b_1,b_2,\dotsc,b_n)\in R^n|\sum\frac{b_i}{i}=0\}$$ Then prove that $S+X$ is a closed set in $R^n$.
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A basic question on symmetry of metric space

In the metric space definiton, the second condition for a metric i.e. symmetry (d(p,q)=d(q,p)) is present. But, I have not seen any example where this condition has been used. Can anyone give any such ...
4
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5answers
721 views

Question on compact metric space.

Is the set of rationals in $[0,1]$ compact? I seems like every open covering should have a finite sub-covering, yet I have read that compact metric spaces are complete...
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0answers
48 views

Does this have a name (metric space related measure of closeness)?

Consider a metric space $(M,d)$, and let $D: M \times M \to \mathbb{R}_+$ be a measure of similarity on it, so that $D(x,y)$ is large when $x$ and $y$ are close (i.e., $d(x,y)$ is small). Consider a ...
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3answers
184 views

Prove $B(x,\delta)$ is open. [duplicate]

Prove $B(x,\delta)$ is open. What this question is asking me to prove...? I don't understand nor have a clue to approach the question...
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1answer
50 views

Confusion regarding the definition of uniform continuous functions on metric spaces.

What exactly is a uniform continuous function on a metric space? My book says $f:X\to Y$ is uniform continuous if $\forall \epsilon\in\Bbb{R}$, for any points $x,y\in X$, there exists a constant ...