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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How does this follow from the Baire category theorem?

The book says that statement 2 is a direct consequence of statement 1. I don't see how they prove statement 2 directly from statement 1, can you please help me? Statement 1: A complete metric ...
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10 views

A connected locally compact metric space is sigma-compact without AC

Is it possible to prove that a connected locally compact metric space is sigma-compact without using the Axiome of Choice? Thank you.
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35 views

Is the metric ${d(x,y)}\over {1+d(x,y)}$ complete where $d$ is the usual Euclidean metric on $\mathbb R^{2}$

Let $d(x,y)$ be the usual Euclidean metric on $\mathbb R^{2}.$ $\mathbb R^{2}$ is complete under $d(x,y)$. I have this subspace given $$[0,1]\times [0,\infty )\ \ of\ \ \mathbb ...
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2answers
22 views

Uniform convergence of Lipschitz functions to characteristic function of a compact set

Consider $(X,d)$ a metric space and $K \subseteq X$ a compact subset. I am trying to build a sequence of Lipschitz functions $f_n : X \to \mathbb R$ s.t. $f_n \to \chi_K$ uniformly. If we try to ...
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3answers
30 views

On the existence of connected metric spaces with open balls not connected

Does there exist a connected metric space with more than one point such that it has an open ball which is not connected ? Moreover does there exist a connected metric space with more than one point ...
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16 views

Proving that S is a metric space

I have the following problem: Let S be the set of bounded functions on [a,b] with $d(x,y) = Sup_{a\leq t \leq b} |x(t)-y(t)|$. Show that S is a metric space. I think that non-negativity and ...
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2answers
39 views

Examples of Incomplete Spaces [on hold]

A metric space is complete if every cauchy sequence is convergent. To make space incomplete either i can change the metric or the ambient space. For example if I change real numbers into rational ...
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Length metric and edge-path metric on a finite dimesional $CAT(0)$ cube complex are coarsely equivalent

I'm trying to find a proof for the statement in the title: Length metric and edge-path metric on the vertex set of a finite dimensional $CAT(0)$ cube complex are coarsely equivalent. Length ...
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44 views

Borel sets: alternative characterization for metric space

For any topological space $(X,\tau)$, the Borel $\sigma$-algebra $\mathcal{B}$ is the $\sigma$-algebra generated by the open sets. In other words, it is the intersection of all $\sigma$-algebras on ...
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1answer
45 views

Understanding Rudin's proof that compact subsets of metric spaces are closed.

Rudin's Principles of Mathematical Analysis has the following definition of compact: A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. ...
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40 views

Show that the image of Lipschitz function $\gamma : [0,1] \to R^n$ has measure $0$, if $n \ge 2$.

Problem Statement: Let $\Gamma$ be the image of a Lipschitz continuous function $\gamma : [0,1] \to R^n$, that is, $\Gamma = \{\gamma(t) : t \in [0,1]\}$, and $|\gamma(t_1) - \gamma(t_2)| \le K |t_1 - ...
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7answers
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What does it REALLY mean for a metric space to be compact? [duplicate]

I've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset ...
2
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1answer
49 views

Is the extended real line a metric space?

I've got a question reading the demonstration of the Theorem 3.2 in POMA of Rudin. Indeed, he says that every convergent sequence in a metric space is bounded. My question is: Is ...
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1answer
14 views

What can we do on $S$ in order that $H(S)$ be compact?

Let be $S$ a metric space. We define the hyperspace $H(S)$ as the metric spaces consisting of every no empty compact subset of $S$ and the Hausdorff metric. I want that $H(S)$ be compact imposing ...
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1answer
24 views

Distance between two ordered sets

Is there a way to measure the "distance" between two ordered sets? Say i got two sequences of letters: $$ S_1 \{A, B, C, D, E, F\} $$ $$ S_2 \{B, C, D, A, F, E\} $$ How could I find an "amount of ...
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23 views

Completeness of metric, normed and inner product spaces

For a metric space to be complete, it needs to have all cauchy sequences converge in the metric. 1) For a normed space to be complete, does it need Cauchy sequences to converge in the norm or in the ...
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1answer
11 views

Determining shapes in metric spaces?

I have a specific and a general question. My specific question is this: how would I determine the shape and location of the set of points satisfying $d(x,a) \leq 1$ in the metric space $(\mathbb{R}^2, ...
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49 views

What is the uniform metric on $\mathbb{R}^X$?

I've been going through Munkres' Topology on my own, and I've come across an exercise where I can't even understand the question. It is exercise 7 of section 21 (p.134): Let $X$ be a set, and let ...
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1answer
31 views

Showing that $\mathbb{R}$ is a metric space with the metric $d(x,y)=|x-y|$

I want to show that $\mathbb{R}$ is a metric space with the metric $d(x,y)=|x-y|$. So three properties of the metric space $d(x,y)$ in general needs to be satisfied. My work: Let $x,y \in ...
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0answers
34 views

Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
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2answers
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Why is the metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ important?

The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ ...
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25 views

elementary problem on uniform continuity in metric space [on hold]

Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be the function defined by $f(x_1,x_2,\ldots,x_n)=\max\{|x_1|,|x_2|,\ldots,|x_n|\}$. Show that $f$ is uniformly continuous.how can i prove its a lipschitz ...
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If $X$ is compact and $C$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number.

Prove the following statement. If $X$ is compact and $C = \{U_\alpha : \alpha \in A\}$ is an open finite cover of $X$ then $C$ has a maximum Lebesgue number. Is my proof correct? Proof: Let $E$ be ...
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bijective uniformly continuous function from a subset of the Cantor set $K$ onto $X.$

Suppose $(X, d)$ is a non-empty metric space. Then $X$ is totally bounded if, and only if, there exists a bijective uniformly continuous function from a subset of the Cantor set $K$ onto $X.$ Proof: ...
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29 views

Topology of Metric spaces. [closed]

I want to be clear about the concept of interior points, boundary points,limit points of a set in metric space. So, I want an explanation with examples on metric spaces. Thank You.
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68 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
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22 views

Strongly Equivalent metrics [closed]

How to show any two metrics to be strongly equivalent? Please suggest me the proper way to show this. Also i want to know how to find the constants in the respective definition.
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0answers
48 views

Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$

Let $M = (\mathbb{R}^2, d_e)$ be the metric space, with $d_e$ the Euclidean metric. Let $C \subseteq \mathbb{R}^2$ be defined by $$C = ([-1, 1]\times[-1, 1]) \setminus \{ y \in \mathbb{R}^2 : d((0, ...
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10 points outside a unit circle

Let $P_1$, $P_2$,... $P_{10}$ be ten points outside the unit circle centered at the origin $O$. Given that $\|P_iP_j\|\ge 1/\sqrt{2}$ for all $1\le 1<j\le 10$, find the minimum of the sum of the ...
2
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1answer
38 views

Prove that $1 / \min \{n\in\Bbb N :x_n\ne y_n \}$ is a metric on the set of all sequences of real numbers

Consider the set of all sequences of real numbers.For $x={(x_n)_n}$ and $y={(y_n)_n}$ we define $N(x,y)=\inf \{n\in\Bbb N :x_n\ne y_n,\text{if $x\ne y$} \}$. Now, $$d(x,y)= \begin{cases} 0, ...
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2answers
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Bounded sequence in a metric space

I have a small question when we have a bouded sequence in a metric space; we say that there exists a closed ball $B'$ such that $(x_n)\subset B'$ or just there exist a ball $B$ such that $(x_n)\subset ...
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52 views

$f$ is continuous, $f : X \to X$, $X$ compact, and $f$ has an $\epsilon$-fixed point for each $\epsilon > 0$. Show $f$ has a fixed point.

Problem: Let $f : X \to X$ be a map from a metric space to itself. A point $z \in X$ is a fixed point of $f$ if $f(z) = z$. Let $\epsilon > 0$. A point $w \in X$ is an $\epsilon$-fixed point of $f$ ...
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1answer
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If $X$ is a Polish space, how do we find an equivalent metric under which $X$ is a totally bounded?

According to Stroock and Varadhan, If $X$ is a Polish space, then one can choose an equivalent metric under which the space is totally bounded (see Stroock and Varadhan - Multidimensional diffusion ...
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44 views

Show that $y_n=x_{\phi(n)}$, defines a Cauchy sequence. [closed]

Let $\phi:\mathbb{N}\to\mathbb{N}$, such that $\displaystyle\lim_{n\to\infty}{\phi(n)}=\infty$. If $(x_n)$, is a Cauchy sequence in the metric space $M$, then $y_n=x_{\phi(n)}$, defines a Cauchy ...
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1answer
52 views

Example of Heine-Borel Theorem

For a subset $S$ of Euclidean space $\mathbb R^n$, S is closed and bounded if and only if S is compact (that is, every open cover of $S$ has a finite subcover). I need an example or ...
2
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1answer
46 views

subspace of a metric space

Let $(S,d)$ be a metric space, $\mathcal{S}$ the induced topology. $A\subset S$ a subset. It is easy to see that $A\cap\mathcal{S}=\mathcal{A}$, i.e., the topological subspace on $A$ is the ...
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1answer
36 views

How to determinate whether superset will be open or closed?

Let $M = (X, d)$ and A is closed subset of X, i.e. $A \subseteq X$. $A$ is told to be closed, iff it's complement $X\setminus A$ is open in $M$. But how can we determine, whether superset is open or ...
2
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3answers
54 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
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1answer
22 views

Find a surjective closed mapping that not an open mapping and a surjective open mapping that is not a closed mapping

Find a surjective closed mapping that not an open mapping and a surjective open mapping that is not a closed mapping Attempt: Suppose $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$. We call ...
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2answers
55 views

Show that the mapping $f → f~'$ from $C^1([0 , 1])$ to $C([0 , 1])$ is not continuous.

Let $C^1([0 , 1])$ be the subspace of $C([0 , 1])$ consisting of the functions that have a continuous derivative throughout $[0 , 1]$. Show that the mapping $\Psi:f → f~'$ from $C^1([0 , 1])$ to $C([0 ...
4
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1answer
43 views

What has “$\delta$ Gromov hyperbolic space” got to do with $\mathbb{H}_n$?

The start of section $2$ of this paper, http://homes.cs.washington.edu/~jrl/papers/kl06-neg.pdf defines something called a ``$\delta$ Gromov hyperbolic space". Can someone explain what has this at ...
4
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0answers
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A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
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1answer
28 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and ...
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0answers
28 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
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2answers
116 views

What is the “topology induced by a metric”?

My book gives the following definition: Let $(M,d)$ be a metric space, and let $\mathcal{T}$ be the collection of all subsets of $M$ that are open in the metric space sense... $\mathcal{T}$ is ...
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50 views

Hilbert Cube and Metric Space

Given that $d(x,y)=\sum_{n=1}^{\infty}2^{-n}|x_{n}-y_{n}|$ defines a metric on $H^{\infty}$ where $H^{\infty}$ is the Hilbert Cube, a collection of all real sequence $x=(x_{n})$ with $|x_{n}|\leq 1$ ...
2
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3answers
174 views

Are strongly equivalent metrics mutually complete?

Maybe I'm missing something, but I can't seem to find any references to my exact question. If two metrics, $d_1(x,y)$ and $d_2(x,y)$ are strongly equivalent, then there exists two positive constants, ...
2
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2answers
52 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
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2answers
68 views

How to determine whether those sets are open or closed?

Given those three sets below, A (left), B (center) and C (right), with A, B, C $\subseteq \mathbb{R^2}$, how can I determine, whether they are open or closed in metric space terminology via simplest ...
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1answer
22 views

how to find the locus when distance from the origin is defined as d(x,y) = max { |x|,|y|},d(x,y) =a (where 'a ' is a non zero constant ) [closed]

How to find the locus when distance of any point from the origin is defined as d(x,y) = max {|x| |y|} where d(x,y) = a ( where is a non zero constant) I have a very long list of questions like these ...