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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Zer0-dimensional, countable, 1st countable T1 space is metrizable?

Show that every countable, first countable, zero-dimensional T1 space $X$ is metrizable. I know that T1 space means that all its singletons are closed. Also, zero-dimensional means that $X$ has a ...
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4answers
92 views

Is there any difference between Bounded and Totally bounded?

Is there any difference between bounded and totally bounded? (in a metric space)
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50 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
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generalization of Banach fixed-point theorem on short maps?

If $ \ T:X \longrightarrow X \ $ is contraction, then using Banach fixed-point theorem we know that the fixed point exists and all other points converge to that point. But what happens if $T$ is not ...
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45 views

Continuous functions on a closed subset of a topological space

Let $X$ be a topological space with $Y$ a closed subspace with relative topology. If $f:Y \rightarrow Z$ is a continuous map of topological spaces, then can $f$ always be extended to be from $X$ to ...
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Show the following extension is lipschitz

$X=S\cup\{x_0\}$, $f:S\rightarrow \mathbb R$ s.t. $|f(s)-f(t)|\leq kd(s,t)$ for $s,t\in X, k>0$. Suppose $x,y \in X$ s.t. $x\in S$ and $y\notin X$then $x=t, t\in S$ and $y=x_0$. I'm trying to ...
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45 views

Strange Property of Ultrametric Spaces and Metric Completion

The following property of ultrametric spaces seems quite strange: (No new values of the metric after completion) Let $x_1, x_2, \ldots$ be a sequence in $X$ converging to $x \in X$. Suppose $a \in ...
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2answers
55 views

What's wrong in this reasoning of $l_\infty$ separability?

While solving a problem, related to functional analysis, I've accidentally got a "proof" of $l_\infty$ being separable, tried to find a fault in it (as the result isn't true), but didn't succeed in ...
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4answers
72 views

closed ball in general metric space?

Is it true that in general complete metric space $(M,d)$, a closed ball of radius $r$ centered at $p\in M$ is always compact? That is, the ball is the set of all points $\left\{x:d(x,p)\leq ...
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1answer
32 views

Cauchy sequence of functions and uniform convergence

If $\Omega$ is a bounded domain, and on $C(\bar{\Omega})$ we use the uniform distance $$d(f,g)=\max_{\bar{\Omega}} |f-g|,$$ a Cauchy sequence of functions (w.r.t. the distance $d$) converge and the ...
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1answer
23 views

Example of a sequence with a Cauchy subsequence and terms arbitrarily close

Let $(X,d)$ be a metric space. Suppose there is a sequence $(x_n)$ in $X$ such that $(x_{2n})$ is Cauchy and $d(x_n,x_{n+1})\to 0$. The question is whether $(x_n)$ is Cauchy? It seems intuitively ...
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10 views

metric space, continuity, open and close

Prove that if F,G are closed in X and f, g are continuous, then f ∧ g is continuous. I know that if I can prove (f ∧ g)^(-1)(A) = f^(-1)(A) ∪ f^(-1)(B), then I know how to prove the rest, can anyone ...
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1answer
22 views

Metric space and compactness

Prove that if in a metric space all closed balls are compact, a subset is compact if and only if it is closed and bounded. Attempt: If all closed balls are compact, then there is a converging ...
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44 views

Does there exist a metric under which $\mathbb{R}$ is incomplete? [duplicate]

Does there exist a metric under which $\mathbb{R}$ is incomplete?
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23 views

compactness and seq.compact of metric space

prove that a sequentially compact subspace of a metric space $X$ is closed in $X$? I wil solve this question from defination of sequentially compact but I dont know how?.(I dont want to solve it from ...
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1answer
37 views

Poincaré inequality on metric spaces

In this book, page 91, example 4.18, it is said that the space $$Y=\{z\in\mathbb{C}:\ |z|=1,\ \arg z\in [-T,T]\}$$ where $3<T<\pi$ does not support a strong Poincare inequality (see page 84 for ...
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90 views

modern analysis: metric spaces and $\varepsilon$-neighborhoods

Prove or disprove that $d(f,g) = ({\int_0^1 |f(x)-g(x)|^{2}dx})^{1/2}$, on $C[0,1]$ is a metric. If so, describe the $\varepsilon$-neighborhood.
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90 views

theorem compactness and Hausdorff

I have this theorem "$X$ is compact $\leftrightarrow\exp X$ is compact", but i can not find source of it. It concerns Hausdorff metric.
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63 views

Metric on the line (1-Dimensional space)

Is the Euclidean metric the only possible metric in real line? Recall that the Euclidean metric is given by $d(x,y) = |y-x|$, and if there is another metric, are these two equal? what about the metric ...
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16 views

Relation between positive definite metric and full basis of a given operator

Let's have some linear space with given indefinite metric. How the fact that metric isn't positive definite is connected with the fact that hermitian (due to the definition of hermicity in a given ...
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30 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
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4answers
71 views

Completeness proof?

First of all, this is not a question about a specific problem, but more about a general technique. When I face a problem such as "show that a metric space $(M,d)$ is complete", the first thing I do is ...
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82 views

Is there a continuous, strictly increasing function $f: [0,\infty)\to [0,\infty)$ with $f(0) = 0$ such that $\tilde d = f\circ d$ is not a metric?

Is there a continuous, strictly increasing function $f \colon [0,\infty)→ [0,\infty)$ with $f(0) = 0$ such that $\tilde d = f\circ d$ is not a metric? You may take $(X,d)$ to be $\mathbb R$ with the ...
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1answer
35 views

Links between Minkowski metric, Hamming distance and Levenshtein distance

I know that Hamming distance is a particular case of Minkowski metric (with the specific definition of the subtraction). Also it seems that Hamming distance is a particular case of a Levenshtein ...
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22 views

Convergence of difference of two cauchy sequence

Let $(X,d)$ be metric space, not complete, and $x_n , y_n$ be Cauchy Sequences in $X$. Then is $d(x_n,y_n)$ convergent? I know that $d(x_n,y_n) \leq d(x_n,x_m)+d(x_m,y_m) + d(y_m,y_n)$, so it is ...
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1answer
48 views

Speed calculation with Diameter and RPM

Hi I am working on a robotics project and I need help with Distance calculation. The diameter of the wheels are $7$ cm or $75$ mm , and the speed on the wheels are constant at all the time whenever ...
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2answers
23 views

Space of probability measures total bounded?

I want to consider a space of probability measures on some set $\Omega$. It's complete (am I right?). But I don't know whether it's total bounded. Actually, I want to prove that the space of ...
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54 views

sufficient conditions that a function has a fixed point

Let be $(X,d)$ a complete metric space and $f:X\to X$ with $d(f(x),f(y))<d(x,y)$. I want to show that in general $f$ has no fixed point. But if $(X,d)$ is a compact space, indeed $f$ has a ...
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1answer
43 views

Covering of closed unit ball with closed balls.

Notations and definitions Let $E$ be a finite dimensional vector space with norm $||\;||$. Let $B$ denote the closed unit ball in $E$ and $B_r[a]$ the closed ball centered at $a$ with radius $r$. ...
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Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
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The distance between an element and a subset of a metric space.

I got stuck on an assignment. Can you help me to solve this? Let $(X,d)$ be a metric space, and let $C$ be a subst. Define the function: $$ f \quad : \quad X \longrightarrow \mathbb{R} \quad : ...
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34 views

Metric Space Properties

I am revising my notes on metric spaces and one of the Theorems is stated but the proof has been ommitted and was looking for some help as I don't know where to start. Let $(X,d)$ be a metric space, ...
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How can the y-axis in $\mathbb{R^2}$ be open?

I have read that $\{(x, \frac{1}{x}): x \neq 0\}$ is closed in $\mathbb{R^2}$. So hence the complement of this set, $\{x = 0\}$, i.e. the y-axis must be open? But we cannot put an open ball with ...
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1answer
71 views

Completeness and separability of $B(X)$ (bounded linear operators on $X$)

Assume $X$ is a separable Banach space with norm $|| \cdot||$. Consider $\{f_n\}_{n \in N}$ a countable dense subset of $X$ and equip $B(X)$ (bounded linear operators on $X$) with the following ...
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1answer
56 views

In what metric spaces is a closed and bounded set compact?

Is there a characterization of a metric space $X$ such that for every $A\subseteq X$, $A$ is compact iff $A$ is closed and bounded? Something that generalizes $\mathbb R^n$?
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1answer
41 views

Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a ...
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45 views

In a metric space, if $A$ is open and $B$ is closed, is $A + B$ open or closed?

Let $A, B \in E^n$, and consider their sum $A + B = \{x+y \mid x \in A, y \in B\}$. Suppose that $A$ is open and $B$ is closed. Is it always true that $A+B$ is open? Is it always true that $A+B$ is ...
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Show that $A$ is closed in $X$ and $f(A)$ is not closed in $Y$.

Let $X=[0,1)$ with the metric $d(x,y)=|x-y|$, and $Y=\mathbb{R}^2$ with the Euclidean metric. Define the mapping $f:X\rightarrow{Y}$ by $f(t)=(cos(2 \pi t + \frac{\pi}{2}), sin(2 \pi t + ...
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Is $\mathbb R^{\omega}$ homeomorphic to $\mathbb R^{\omega} \times \mathbb R^{\omega}$?

As a study exercise, I'm trying to find a topological space $X$ which is homeomorphic to $X \times X$. I began thinking of simple examples involving $\mathbb R$ but then realized my best bet would be ...
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1answer
26 views

Is a finite subset of a complete metric space again a complete metric space?

The space $(\mathbb{R}^2, d)$ where $d(x,y)=max \{|x_1-x_2|, |y_1, y_2|\}$ for $x=(x_1, y_1)$ and $y=(x_2, y_2)$ $\in X$ is a complete metric space. Let $X=\{(0,0), (-\frac{1}{8}, 0), (0, ...
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connection of cond. neg. def. of metric $d(x,y)$ and pos. semidef. of kernel $k(d(x,y))$?

I have the following problem: I've given a conditionally negative definite metric of the form $d(x,y):=|x-y|^\alpha$, and a kernel function $k(x,y):=Ed(x,X)+Ed(y,X)-Ed(X,\tilde{X})-d(x,y)$. ...
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53 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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52 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
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30 views

prove the set of all spheres with rational center and radius is countable

Prove the set of all spheres in $\mathbb{R}^3$ with rational center and radius is countable. I have two ideas. Is either one better than the other? 1) let $(x,r)$ represent a sphere in ...
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29 views

$A$ and $B$ compact in a hausdorff space implies $A\cap B$ is compact

Prove that if $A$ and $B$ are compact subset of a hausdorff space $X$, then $A$$\cap$$B$ is compact.
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Continuity of identity in $p$-adic $\mathbb Z$

Say we have the $p$-adic metric in $\mathbb Z$ defined as $$ d_p(a,b)= \left\{\begin{align} &0 & a=b \\ &p^{-r} : p^r\mid (a-b), p^{r+1}\nmid (a-b) & a\neq b \end{align}\right. $$ I'd ...
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36 views

Completeness & Closedness in Metric Spaces

If every (proper) closed subset of a metric space is complete, then is the whole space necessarily complete as well?
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What is this space with infinitely many different points with distance $1$ between any two different points?

I'm reading Mac Lane's: Mathematics, Form and Function: [...] There are also bizarre examples - such as "a space" with infinitely many different points, with distance $1$ between any two different ...
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Sequence Criterion of Continuity and Divergent sequences

A function $f : (X, d_X) \to (Y, d_Y)$ between metric spaces is continuous iff $$ \lim_{n \to \infty} f(x_n) = f(\lim_{n\to \infty} x_n) $$ for each convergent sequence $(x_n)$ in $(X, d_X)$. So if I ...
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Proving that a set of metric space is dense in $A$ iff there exists a sequence converging to $x\in A$

I'm using the following definition: A set $M$ of a metric space $(\frak M,\rho)$ is called dense in a set $A\subset\frak M$ if $$\forall \varepsilon>0,x\in A\exists y\in ...