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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540 views

What is a metric for $\mathbb Q$ in the lower limit topology?

A useful source of counterexamples in topology is $\mathbb R_\ell$, the set $\mathbb R$ together with the lower limit topology generated by half-open intervals of the form $[a,b)$. For example this ...
2
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1answer
133 views

Generalization of metric spaces

The Wikipedia article for metrics mentions several generalizations of metric spaces, but all of them seem to have the property that the metric must be non-negative for all x and y. To me it seems like ...
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1answer
927 views

Proving that a metric space is compact

Let $H^\infty$ be the set of real sequences such that each element in each sequence has $|a_n|\leq 1$. The metric is defined as $$d(\{a_n\}, \{b_n\}) = \sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n}.$$ ...
5
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3answers
192 views

The metrizable space may be not locally compact

My text book said: Not every metrizable space is locally compact. And it lists a counterexample as following: The subspace $Q=\{r: r=\frac pq; p,q \in Z\}$ of $R$ with usual topology, i.e., ...
0
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1answer
193 views

variation problem of constrained area and minimized distance

$$c=\int_{x_1}^{x_2}f_{gr}(x)\;dx$$ The integral is a time-like curve between $x_1$ and $x_2$ and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and ...
1
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1answer
175 views

Uniform convergence of functions, Spring 2002

The question I have in mind is (see here, page 60, the solution is at page 297): Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and ...
6
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2answers
650 views

$p$-adic completion of integers

I'm trying to do the following exercise: Let $p$ be a prime and for $n\geq 1$ let $\alpha_n :\mathbb Z/p \mathbb Z \to \mathbb Z/p^n \mathbb Z$ be the injection of abelian groups given by $1 \mapsto ...
4
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3answers
470 views

Cauchy sequences in metric spaces

Let $(X,d)$ be a metric space and let $(x_n)_{n\in\mathbb{N}}$ be a Cauchy sequence in $X$, i.e. $d(x_n,x_m)$ goes to $0$ when $n,m\rightarrow\infty$. The sequence does not necessarily have a limit in ...
1
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1answer
302 views

Finding a convergent subsequence of a sequence of functions in a metric space.

This question has some context, which I explain below. Let $(X, d)$ be a metric space, and fix some $x_0$ in $X$. Define $$BC(X) = \{ f : X \rightarrow \mathbb{R} : f~\text{is continuous and ...
3
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1answer
174 views

Counterexample of Compactness

Let $X$ be a metric space and $E\subset X$. Let {$G_i$} be an open cover of $E$ For every open cover {$G_i$}, there exists a finite subcover {$G_{i_n}$} of $E$ such that $G_{i_n} \in${$G_i$}. For ...
3
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2answers
233 views

If a subset of $\mathbb{R}$ is closed and bounded with respect to a metric equivalent to the Euclidean metric, must it be compact?

Two different metrics $d$ and $\hat d$ in a space $X$ are said to be equivalent iff the topologies generated by them are the same, in other words $U\subseteq X$ is $d$-open iff it is $\hat d$-open. ...
4
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1answer
232 views

Is this metric space complete?

No, it is not complete metric space: by Stone-Weierstrass thm we know that $|x|$ can be uniformly approximated by sequence of polynomials which are clearly $\mathcal{C}^1[0,1]$, but $|x|$ is not ...
4
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3answers
535 views

Separability of a product metric space

I am trying to prove the following: 'If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is ...
3
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1answer
239 views

Topological equivalence between $\Bbb R^n$ and itself.

To start with, let me just state this theorem: THEOREM 1 Let $(X_i,d_i)$,$(Y_i,d_i^{\,\prime})$ be metric spaces for $i=1,\dots,n$. Let $f_i:X_i\to Y_i$ be continuous for each $i$ as functions ...
4
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3answers
1k views

Difference between Norm and Distance

I'm now studying metric space. Here, I don't understand why definitions of distance and norm in euclidean space are repectively given in my book. I understand the difference between two concepts when ...
2
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1answer
137 views

Convergence of a function in the continuous functions metric space with infinite norm induced metric.

I know that $(C[0, 2], d_{\infty})$ is a complete metric space, being $C[0, 2]$ the set of continuous functions in the closed interval $[0, 2]$ and $d_\infty$ the distance metric induced by the ...
20
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2answers
340 views

A strangely connected subset of $\Bbb R^2$

Let $S\subset{\Bbb R}^2$ (or any metric space, but we'll stick with $\Bbb R^2$) and let $x\in S$. Suppose that all sufficiently small circles centered at $x$ intersect $S$ at exactly $n$ points; if ...
4
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2answers
758 views

$d(x,A)=0\iff $ every neighborhood of $X$ contains a point of $A$

Mendelson, Introduction to Topology, p.52 $(8)$. Let $A$ be a non-empty subset of a metric space $(X,d)$. Let $x\in X$. Prove that $d(x,A)=0$ if, and only if, every nieghborhood $V$ of $x$ ...
4
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1answer
111 views

Limit preserving metrics.

I need to prove that given a sequence of points $\{a_n\}$ in $(\Bbb R^k,d_m)$ that converges to $a$, then it converges to the same limit in both $(\Bbb R^k,d_e)$ and $(\Bbb R^k,d_t)$ (and conversely), ...
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1answer
3k views

A isometric map in metric space is surjective? [duplicate]

Possible Duplicate: Isometries of $\mathbb{R}^n$ Let $X$ be a compact metric space and $f$ be an isometric map from $X$ to $X$. Prove $f$ is a surjective map.
3
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3answers
312 views

Understanding compact subsets of metric spaces

Please help me understand the following definition: Let $(X,d)$ be a metric space, a subset $S \in X$ is called compact, if any infinite sequence $\{x_{n}\}_{n\in\Bbb N}\in S$ has a sub-sequence ...
2
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1answer
94 views

Balls and transformed sets in normed vector spaces

Let $T$ be a surjective, continuous linear operator between two Banach spaces $E$ and $F$. Assume that it is $B_F(y_0,4c)\subset \overline{T(B_E(0,1))}$, where $c>0$, $y_0 \in F$ ($B$ is for ...
4
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2answers
536 views

which of the following metric spaces are separable?

which of the following metric spaces are separable? $C[0,1]$ with usual 'sup norm' metric. the space $l_1$ of all absolutely convergent real sequences, with the metric ...
3
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1answer
1k views

Continuous extensions of continuous functions on dense subspaces

I thought that if I have a function $f: \mathbb Q \to \mathbb R$ that is continuous then I can (uniquely) extend it to a continuous function $F: \mathbb R \to \mathbb R$ as follows: for $r \in \mathbb ...
4
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1answer
471 views

which of the following metric spaces are complete?

Which of the following metric spaces are complete? $X_1=(0,1), d(x,y)=|\tan x-\tan y|$ $X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$ $X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$ $X_4=\mathbb{R}, ...
5
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3answers
244 views

Further reading on the $p$-adic metric and related theory.

In his book Introduction to Topology, Bert Mendelson asks to prove that $$(\Bbb Z,d_p)$$ is a metric space, where $p$ is a fixed prime and $$d_p(m,n)=\begin{cases} 0 \;,\text{ if }m=n \cr ...
48
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1answer
919 views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
1
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1answer
177 views

Maximal color difference

I have a picture consisting of a two-dimensional array of ordered triples (red, green, blue) of real numbers from 0 to 1. I'm looking for something like a norm on pictures which expresses the range of ...
3
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1answer
59 views

Relation between metrics

Let $$\eqalign{ & d\left( {x,y} \right) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} \left\{ {\left| {{x_i} - {y_i}} \right|} \right\} \cr & d'\left( {x,y} \right) = \sqrt ...
2
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1answer
207 views

Distance between a sequences and compact sets

Let $(X,d)$ be a metric space, and $K\subset X$ compact. Define for $x\in X$, $$\rho(x, K)=\inf_{y\in K}d(x,y).$$ Let $(x_n)_n\subset X$ be a sequence in $X$ such that $\rho(x_n,K)\to 0$. Is it true ...
6
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5answers
252 views

Is a countable subspace of a metric space closed?

Let $(X,d)$ be a metric space and $a_1,a_2,\ldots\in X$. Define $A=\{a_n:n\in\mathbf N\}$. Is $A$ closed in $(X,d)$? And is $A$ closed when X is a topological space?
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2answers
479 views

Index notation and differentiation

Let $x_i$ such that $i=1,2,\ldots,n$, and $\vec{x}=(x_1,\ldots,x_n)$ Define $$A:= M_{ij}(\vec{x})\dot{x}^i\dot{x}^j$$ where Einstein summation applies. Also, $M$ is symmetric and invertible -- a ...
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2answers
162 views

How to show $\alpha(A)\leq \beta(A)\leq 2\alpha(A)$

Let $X$ be a metric space and let $A\subset X$ be a bounded subset of $X$. I read on Wikipedia that the Hausdorff- and Kuratowski measures of non-compactness ($\alpha$, resp. $\beta$) satisfy the ...
1
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1answer
151 views

How to show this property for the diameter of a set

Let $(M, d)$ be a metric space and $A\subset M$ such that $$\mathrm{diam}(A)=\sup_{a,b\in A}d(a,b)=D<\infty.$$ How can I prove that for any $\varepsilon> 0$ there is $x\in A$ such that ...
2
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1answer
66 views

Hölderian path connectedness

Let $X$ be a complete metric space and $\alpha \in (0,1)$. Suppose that for every $x,y \in X$ there exists $z \in X$ s.t. $$ d(x,z) \le \frac{1}{2^\alpha}d(x,y), \qquad d(y,z) \le ...
2
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1answer
411 views

Converse of “a convergent sequence is Cauchy”

I know that a metric space $X$ is called complete if and only if every Cauchy sequence in $X$ converges to a point in $X$. But we can also say something about Cauchy sequences in incomplete spaces. ...
5
votes
2answers
259 views

Proving a metric space to be compact

I have the following metric space: The set $X$ of all sequences with members from the set $\{1,2,\ldots, n\}$, together with the metric $$d(x,y)=\frac{1}{\min\{j\in\mathbb{N}:x_j\ne y_j\}}.$$ I wish ...
2
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1answer
824 views

Intersection of countable set of compact sets

I am asking whether a specific construction is a counterexample to Theorem 2.36 in Rudin's "Principles..." book (3rd Ed.), which reads, 2.36 Theorem If $\{K_{\alpha}\}$ is a collection of compact ...
2
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1answer
85 views

A question about weakening the conditions of Schauder's fixed point theorem

I'm currently doing a course on the theory of metric spaces. This is the version of Schauder fixed point theorem from my course: Let $(X,\|\cdot\|)$ be Banach and $C\subset X$ a closed, bounded, ...
6
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1answer
213 views

Extending isometries between compact subspaces of Cantor space

Let $\omega$ be the set of natural numbers. $2^\omega$ is the Cantor space. Suppose $K$, $L \subset 2^\omega$ are compact, and there is an isometry $f: K \to L$. Then how could one extend $f$ to an ...
2
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0answers
108 views

What is the correct distance measure for the (anti) de-Sitter space?

Given these two expressions 1) $\sinh{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1−(t^2−x^2)}}$ 2) $\sin{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1+(t^2−x^2)}}$ for distance $d$ from the origin $(0,0)$ to point $(x,t)$, ...
22
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4answers
4k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
1
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1answer
70 views

Something weaker than the Riesz basis

I have some function $f$, real valued and continuous. I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in ...
-1
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1answer
122 views

Practical implications of a vector space being a topological vector space

I have a space $V$ and I lately discovered that it's a topological vector space. What are the practical implications of that?
0
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1answer
254 views

Confusion about Completion of Metric Space Proof

I have started studying Functional Analysis following "Introduction to Functional Analysis with Applications". In chapter 1-6 there is the following proof For any metric space $X$, there is a ...
3
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1answer
118 views

The Limit of a Sequence of Paths

Given a path connected topological space $X$, consider a sequence $x_1, x_2, x_3, \dotsc$ of points in $X$ converging at some $x \in X$. For each $x_i$ and $x_{i+1}$, there exists some path $p_i$ in ...
5
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2answers
526 views

If $f_n\colon [0, 1] \to [0, 1]$ are nondecreasing and $\{f_n\}$ converges pointwise to a continuous $f$, then the convergence is uniform

Suppose that $\{f_n\}$ is a sequence of nondecreasing functions which map the unit interval into itself. Suppose that $$\lim_{n\rightarrow \infty} f_n(x)=f(x)$$ pointwise and that $f$ is a continuous ...
3
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2answers
82 views

Entropy of a North South Transformation.

Let $f:\mathbb{S}^2\to\mathbb{S}^2$ be a continuous north south Transformation, in other words, the point $(0,0,1)$ is a global attractor for $f$ and $(0,0,-1)$ is a global attractor for $f^{-1}$. ...
3
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1answer
175 views

Completeness and separability of Lévy's metric

Let $D$ be the set of all functions $F: \mathbb{R} \rightarrow \mathbb{R}$ which are nondecreasing, left-hand-side continuous and $\lim_{x \rightarrow -\infty} F(x)=0$ and $\lim_{x \rightarrow ...
1
vote
1answer
300 views

Levy-Prokhorov metric question

I have a question relating to the Lévy-Prokhorov metric and its description on wikipedia. The metric is defined for measures on $\mathbb R^d$ and is defined by $$ ...