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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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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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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0answers
6 views

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 ...
0
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1answer
32 views

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 ...
3
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2answers
42 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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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 ...
2
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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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1answer
38 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 - ...
9
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7answers
495 views

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 ...
0
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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 ...
2
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1answer
48 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
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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0answers
22 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
38 views

Using CAT(0) inequality

Let $X$ be a CAT(0) space with metric $d$. Let $p,x,y$ three points on $X$, and let $u,v$ be points on geodesic $[p,x]$ and geodesic $[p,y]$ such that $d(p,u)\geq a,d(p,v)\geq a$,where a is some ...
2
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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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0answers
22 views

Find the $\epsilon - \delta$ values for the continuous function - modified step function defined on $[0,1]$

Let the modified step function be defined on $[0,1]$ by : $f(x) = \begin{cases} \bigg( \dfrac {2^n+1}{2}\bigg )x - \dfrac {2^n-1}{2^n} ; & n \in \mathbb N~~ , \dfrac {2} {2^n+1} ...
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1answer
68 views

Implications of inner products vs normed spaces vs metric spaces

Is it true that: -an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality -a norm can be induced by a metric if and only if the metric ...
0
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0answers
29 views

munkres topology the meaning of 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's in section 21, number 7. Let $X$ be a set, and let $f_n: ...
2
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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 ...
2
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0answers
33 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}, ...
3
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2answers
93 views

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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1answer
33 views

Closed subspace of a metric space in which distance between any two points is at most $1$

$X$ be a metric space and $Y$ be a closed subspace of $X$ such that distance between any two points is at most $1$. Then $1$. $Y$ is compact $2$. Any continuous function from ...
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2answers
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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0answers
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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1answer
309 views

Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
3
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1answer
64 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
3
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1answer
35 views

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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0answers
46 views

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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13 views

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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0answers
29 views

Topology of Metric spaces. [on hold]

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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0answers
22 views

Strongly Equivalent metrics [on hold]

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.
4
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0answers
66 views

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: ...
2
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2answers
68 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
3
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2answers
77 views

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 ...
1
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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, ...
2
votes
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
14 views

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 ...
3
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0answers
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 $f$ ...
0
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1answer
35 views

Book recomendation for function sequences.

I wanted to study about sequences of functions defined in metric spaces. What book/books do you recommend? Thanks!
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3answers
41 views

In a normed vector space, if $O $ is an open set then $ O+a $ is open?

In a normed vector space, if $O $ is an open set then $ O+a $ is open? Here $ a$ is an element of some other set $ A $ . This feels intuitively obvious, as we just have "moved" the entire set, but I ...
0
votes
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 ...
4
votes
2answers
171 views

Trying to show that $C([0,1])$ is a complete metric space, using the norm $\|f|| = \max_{x\in [0,1]} |f(x)|$.

I think I have this problem almost done. I am taking $C([0,1])$ to be the set of all continuous function $f\colon[0,1] \to \mathbb{R}$. I have already shown that $\displaystyle\|f\| = \max_{x\in ...
4
votes
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 ...
2
votes
3answers
51 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 ...
2
votes
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 ...
0
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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 ...
0
votes
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 ...
1
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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 ...
0
votes
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 ...
-3
votes
0answers
49 views

about cauchy sequence in metric space [closed]

Let $f$ be a function from a metric space $(X,d_1)$ to a metric space $(Y,d_2)$. If the image of every Cauchy sequence in $X$ is a Cauchy sequence in $Y$, how can I prove that $f$ is continuous?