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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Proving the metric attains a minimum on a compact subset

Let $(X,d)$ be a complete metric space. Suppose $B \subset X$ is compact. Prove that for every $a\in X$ the minimum $\min_{b\in B} d(a,b)$ exists. I'm pretty sure you can do this by just using the ...
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49 views

Connected $G_\delta$ sets in a connected completely metrizable space with more than one point.

Suppose $(X,\tau)$ is a connected completely metrizable space with more than one point. Let $\mathbb{G}$ be the set of all connected $G_\delta$ subsets of $X$. And let $\mathbb{O}$ be the class of ...
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In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
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1answer
23 views

Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
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61 views

Do connected complete metric spaces always contain a path?

Does every connected complete metric space with more than one point contain a non-trivial path? The pseudo-arc is an example of a connected metrizable space without a path.
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36 views

Connected complete metric spaces with more than one point.

Does every connected complete metric space with more than one point have infinitely many closed balls? And is any closed ball in a connected complete metric space connected?
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51 views

if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.
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2answers
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Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
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1answer
26 views

Function vs. Polynomial Space

I've been reading up on spaces and was wondering if there was a difference between those two terms? Intuitively it would seem they are the same, but just so I don't dig myself into a hole, I was ...
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30 views

Notation problem with a set of tuples and a metric

The first question: Assume we have tuples $T_i = (x_i, \vec{c}_i)$ ($x_i$ is the name of the object which is characterized by $\vec{c}_i$ in a d-dimensional space) and define a set of them $TS = ...
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24 views

Lorentz transformation and Minkowski metric

For the exam I'm trying to solve some problems. Today I found this exercise and need some help: For the group S0(1,1) of the Lorentz transformation I have $\phi \in \mathbb{R}$ and $A_{\phi}: ...
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47 views

What is the difference between a Metric Space and a Pseudo-Metric Space?

I was wondering if anyone had information that would help me better understand the difference, so that I can work better on: Interesting Metrics I took a look at Metric assuming the value infinity ...
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43 views

Showing that the minimum distance between a closed and compact set is attained

I have two subsets of $\mathbb{R}^n$, given by $K$ and $F$, $K$ is compact and $F$ is closed. I'm trying to show that $\inf\{ d(x,y) : x \in K, y \in F \}$ is attained. My ideas so far: I know that ...
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2answers
39 views

Is there a meaning to convergence, limits and closedness in pseudo-metric spaces?

A. A sequence ($x_n$) in a metric space $M$ is said to converge to the limit $x \in M$ if the distance between $x_n$ and $x$ converges to 0 as $n$ goes to infinity. What happens when $M$ is a ...
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1answer
59 views

Representative elements in the symmetric difference metric

The symmetric difference is a natural way to quantify the distance between measurable sets: $$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$ This is a pseudo-metric because there may be ...
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1answer
40 views

Is the geometric mean of two metric spaces a metric space?

Suppose there are two metric spaces $d_1$ and $d_2$ over the set $X$. For $x,y \in X$, is $d_3(x,y) =\sqrt{d_1(x,y)d_2(x,y)}$ a metric space? I am having trouble with the triangle inequality. It is ...
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65 views

Prove that this is a metric

$d:\Bbb C \times \Bbb C \to \Bbb R$ Defined by $$d(z,w) := 2\frac{|z-w|}{\sqrt{(1+|z|^2)(1+|w|^2) }},$$ prove that $d$ is metric in $\Bbb C$. I had proved $d$ satisfies the two conditions to be ...
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Isometries from $\mathbb{R}$ to itself.

Prove that every isometry from $\mathbb{R}$ to itself is either $j_a$ or $i \circ j_a$. Here, $j_a$ is defined as $x\mapsto x+a$, and $i$ is defined by $x\mapsto -x$. Also, we're assuming the ...
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1answer
53 views

The real numbers as a completion of the rationals

The real numbers are the completion (i.e. Cauchy sequences modulo equivalence) of the rational numbers. I want to define the real numbers this way, but without using uniform spaces. The problem is ...
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1answer
44 views

How to check whether this function is continuous or not..?

Let A and B be two disjoint closed sets of any Metric space X.I have to construct a continuous function such that $f(x):= 0$ if x belongs to $A$ $f(x) = 1$ if x belongs to $B$ My idea is to use the ...
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2answers
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The completeness assumption in Prokhorov's theorem

Originally, I encountered this question on Terence Tao's blog, where the following exercise is presented: Exercise 23 (Implications and equivalences) Let $X_n, X$ be random variables taking values ...
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Proving compactness of the extended complex plane

Prove that $(\overline{\mathbb C}, \overline{d})$ with $\overline{d}(z,z')=d(\phi(z),\phi(z'))$, where $d$ denotes the euclidean distance in $\mathbb R^3$ and $\phi$ is the inverse of the ...
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1answer
28 views

Showing infimum of distance is attained

I have a continuous map $f: X \to X$ on a compact metric space and I am trying to show that $inf \{ d(x,f(x)) : x \in X \}$ is attained. My thoughts so far are to use sequential compactness to obtain ...
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1answer
52 views

Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
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Interesting Metrics

To preface, this question might come off as a bit silly, but it would serve me well to understand the concepts, and to actually get a good answer for this. How can I design an ideal metric for ...
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Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
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37 views

Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
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Completeness in a Category of Metric Spaces.

Is there a way to describe completeness within a category of metric spaces? The point is that I'd like to have a description of compactness in metric spaces by something of the form totally bounded + ...
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1answer
58 views

Subset of infinite connected set

How to proove that infinite connected set has got proper infinite connected subset?
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43 views

How to prove that a metric space is compact if it is complete and totally bounded?

How to prove that a metric space is compact if it is complete and totally bounded? Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
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Which of the following sets are dense in $C[0,1]$

Which of the following sets are dense in $C[0,1]$ with respect to sup-norm topology? $1$. {$f$$\in$ $C[0,1]$ : $f$ is a polynomial } $2$. {$f$$\in$ $C[0,1]$ :$f(0)$=$0$} $3$. {$f$$\in$ $C[0,1]$ ...
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Is distance between two sets equal to that between their boundary?

I am not sure if the statement below is true. The statement is: Let $(M,d)$ be a connected metric space and $A, B$ be two nonempty subsets of $M.$ Assume the boundary $\partial A$ and $\partial B$ are ...
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1answer
31 views

A question about closed ball in metric space

Question: Let $(M,d)$ be a metric space and $\Omega$ be a bounded open subset of $M.$ For every positive real number $\epsilon,$ let $$\Omega_{\epsilon}:=\{x\in\Omega \mid ...
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1answer
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Is the diameter of intersection of a set with a sphere of radius $r$ a measurable function of $r$?

I have to face to following problem: let $X$ be a separable metric space and $x_0 \in X$ fixed. Consider an open bounded set $A \subset X$. I want to know if the function $f: [0, \infty) \mapsto [0, ...
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1answer
31 views

Can someone criticise my incorrect proof about a set being open?

In the question I have to decide whether the set $S=\{(x,y)\in\mathbb{R}^2\;|\;x/y\leq 7\}$ is open, closed or neither. I attempted to prove it was closed but it turns out it is neither can someone ...
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1answer
16 views

$l_{p}$ metric on $\mathbb{R}^{n}$ and its open balls

For $x,y \in \mathbb R^n$ let $$ d_p(x,y) = \left(\sum_{i=1}^n \def\abs#1{\left|#1\right|}\abs{x_i - y_i}^p\right)^{1/p}$$ for $1 \le p < \infty$ and $$ d_\infty(x,y) = \max\{\abs{x_i -y_i} ...
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triangle inequality to show metric

$d(x,y)= \begin{cases} 0 &\mbox{if } x=y \\ 1+\frac{1}{x+y} & \mbox{if } x\neq y \end{cases} $. Show that $(\mathbb{Z}^+,d)$ is a metric space. I'm stuck in proving triangle inequality.
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Help Me Understand: Proof that Finite Intersection of Open Sets is Open

The proof is here: (link). I don't see how the third line (starting with Thus: $\exists \epsilon_i$...) is justified. That is: just because $x \in U_i$, for all $i$, how do I know that a ...
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2answers
26 views

replace convergence with continuity?(metric spaces)

This question is convcerning metric-spaces. In theory we can replace continuity with convergence. That is, since continuity in a point a is equal to the statement that if $\{x_n\}$ is any sequence ...
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1answer
11 views

A question about equivalent metrics.

I'm trying to prove that if the convergent sequences of $(X,d)$ and $(X,\rho)$ are the same, then the metrics $d$ and $\rho$ are equivalent. Equivalent metrics are those that generate the same open ...
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1answer
16 views

Net-Complete $\iff$ Sequence-Complete

Prove that a metric space is complete w.r.t. sequences iff it is complete w.r.t. nets! (The converse is trivial of course!)
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1answer
43 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
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Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
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Can I show these questions (is a set open or closed WRT metric) a faster way?

I have the metric: $$d((x,y),(a,b))=|y-b|\text{ if }a=x\text{ else }|y|+|b|+|x-a|$$ I have been asked the following questions: Is the set $\{0\}\times(0,1)$ open with respect to this metric? Is it ...
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1answer
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Difference between F-space and Frechet space in W. Rudin's “Functional Analysis”

In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$. In the vector space context, the term local base will ...
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1answer
40 views

int$(A) \subseteq$int$(A')$ and int$(A) \subseteq A'$

In which metric spaces is it true that int$(A) \subseteq$int$(A')$ ? (I know it is true in $\mathbb R$) Moreover in which metric spaces is it true that int$(A) \subseteq A'$ ? (I know it is true in ...
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1answer
21 views

“Every seq. in $X$ has a Cauchy subseq.” implies “$\forall\epsilon > 0, \exists $ a finite set $T$, s.t. $\forall x\in X, d(x,T)<\epsilon$.”

I have a proof of the following theorem. Let X be a metric space. "Every sequence in X has a Cauchy subsequence" implies that "$\forall\epsilon > 0, \exists $ a finite set T, s.t. $\forall x\in X, ...
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1answer
27 views

Function that's a metric on one space but not another?

Is there a function which makes sense on two sets and is a metric on one but not the other? I can't seem to come up with an example or a proof a metric on one set implies it is on every other one it ...
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32 views

For any countable $ A$ , $B \subseteq A \implies B \cap B\space' \ne B $

In which kind of metric spaces is the following true For any non-empty countable set $A$ of the metric space , $B \subseteq A \implies B \cap B\space' \ne B $
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1answer
125 views

Hyperspace and connectedness

I'm looking for any theorems and proofs for connectedness for hyperspaces exp(X). I would like to take a look for especially this theorem: $$ X \textit{ is connected } \leftrightarrow exp(X) ...