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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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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2answers
63 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 ...
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1answer
32 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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49 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
19 views

If $X$ is a polish space, how do we find an equivalent metric under wich $X$ 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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0answers
34 views

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

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
50 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 ...
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1answer
44 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
50 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
20 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
52 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 ...
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1answer
42 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 ...
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0answers
48 views

about cauchy sequence in metric space [on hold]

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?
3
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0answers
52 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: ...
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1answer
27 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
27 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
115 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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49 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
154 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
50 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
67 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 ) [on hold]

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 ...
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1answer
8 views

A point $a=(a_1,…,a_n)$ is isolated point in the cartesian product

Let be $M$ a metric space. A point $a=(a_1,...,a_n)$ is isolated point in the cartesian product $M=M_1\times...\times M_n$, if and only if, each coordinates $a_i$ is a isolated point in $M_i$ My ...
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1answer
17 views

Intersection of a dense set with an open set is dense in the open set

Let $A\subset M$ ann open subset, of the metric space M. If $X\subset M$ is dense in M, then $X\cap A$ is dense in A. My approach: If $X\subset M$, and $A\subset M$ is a open subset. Let be ...
2
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3answers
45 views

When the set of $r$-far interior points from a set is open

Let $E$ be a subset of a metric space $X$ and for $r > 0$ let $$ E_r = \lbrace x \in E : d(x,E^c) > r \rbrace .$$ Is the set $E_r$ always open? Equivalently, is the function $ x \mapsto ...
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1answer
29 views

Prove complete metric space for $I=]0,\infty[$ with $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$ [duplicate]

Let $I=]0,\infty[$ equipped with the metric $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$, $\forall x,y \in I$. Prove that $(I,d)$ is complete. Any help, and thanks in advance.
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1answer
32 views

Topology induced by discrete metrics and topology induced by singleton [closed]

Show that the topology generated by singleton sets is topology induced by discrete metric. $$d(x,y)= \begin{cases} 0,&\text{if } x=y\\ 1,&\text{if } x \ne y\\ \end{cases} $$
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2answers
76 views

Showing that a rectangle is equal to the closure of its interior

I'm trying to show that if Q is a rectangle, then Q equals the closure of Int Q. I have that the closure of Int Q is a subset of Q and I'm now working to show that Q is a subset of the closure of Int ...
2
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2answers
64 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 ...
2
votes
5answers
138 views

A-noncompact, Does there **always** exist a continuous function $f: A \to \mathbb R$ which is bounded but does not assume extreme values?

It's well known that if $ A \subset \mathbb R$ is compact then every continuous function $f:A \to \mathbb R$ is bounded and assume extreme values .So the obvious question is: Given any non compact ...
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1answer
34 views

Example of a separable, locally-compact metric space which is not $\sigma$-compact

I am looking for an example of a separable, locally-compact metric space which is not $\sigma$-compact. At first I thought I could show that if a metric space is separable and locally-compact, then ...
1
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2answers
40 views

Given any non compact set $A \subset \mathbb R^n$ does there exist a continuous function $f: A \to \mathbb R$ which is not uniformly continuous?

It's well known that if $ A \subset \mathbb R^n$ is compact then every continuous function $f:A \to \mathbb R$ is uniformly continuous.So the obvious question is: Given a non compact set $A \subset ...
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0answers
30 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...
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1answer
93 views

Let $f:M\to N$ be continuous, then $f(U)\subset V$.

Let $f:M\to N$ be continuous, with $M$ and $N$ metric spaces. Given an arbitrary subset $X \subset M$ and an open set $V \subset N$, with $f(X) \subset V$, prove that there exists an open subset $U$, ...
8
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65 views

Question on complete metric spaces and whether the following is a complete metric space:

Let $ S \subset C^2([0,1])$(set of all two-times differentiable functions on $[0,1]$), which satisfy $$f(0)+f(\frac{1}{2})+f(1)=0.$$ Question :Is $ (S,d)$ is a complete metric space, where $d$ is ...
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0answers
32 views

Topology Question in Munkres Analysis on Manifolds

Let X be a metric space with metric d. Let Y $\subset$ X. Give an example where A is open in Y but not open in X. Give an example where A is closed in Y but not closed in X. I'm stuck on question ...
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2answers
25 views

What is the difference between 1) given any , 2 ) for every 3) for all in definitions?

I have a lot of confusion. Definition: A sequence $\{x_n\}$ of points in a metric space is said to converge if there is a point $p \in S$ with the following property: For every ...
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1answer
56 views

Prove $d(x,y)=(x-y)²$ is a metric

How can I prove that $d(x,y)=(x-y)²$ is a metric? I have proven all properties except the triangle inequality, i.e $d(x,y) \le d(x,z)+d(z,y)$ I have tried to prove it using the fact that ...
5
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1answer
118 views

Metric limit and limit in category

Is it possible to construct a category $\mathcal{C}$ with $\mathrm{Ob}\,\mathcal{C}=\mathbb{R}$ and for every diagram of the from $$a_0\leftarrow a_1\leftarrow\cdots a_n\leftarrow\cdots$$ the inverse ...
6
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1answer
32 views

A proper subspace of a normed vector space has empty interior.

In a vector normed space $E$, prove that all vectorial subspace $F\neq E$ has a interior empty. My approach:We consider, the open ball $B\subset F$, with $F$ proper subspace of $E$. If $x\notin F$, ...
5
votes
1answer
56 views

Continuity of a “minimal distance” projection $f:(X,d) \to (K, d_{|K})$ for a compact $K \subset X$. (Hint preferred)

Let $(X,d)$ be a metric space and $K$ be a compact subset of $X$. Show that for every $x \in X$ there exists $k_x \in K$ such that $$d(x,K)=d(x,k_x)$$ Suppose that for every $x\in X$, there exists ...
1
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1answer
53 views

Countable choice and totally bounded metric spaces

Can we prove that the following statement is equivalent to the axiom of countable choice (CC)? If every sequence in a metric space $X$ has a Cauchy subsequence, then $X$ is totally bounded. ...
0
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0answers
47 views

Open Set in the Cartesian plane.

I'm trying to prove that the following set is an open set in $\mathbb{R}^2:$ $$A=\{(x_{1},x_{2})\in\mathbb{R}^{2}: x_{1}+x_{2}>1\}$$ with respect to norm $||x||_{1},||x||_{2},||x||_{\infty}.$ ...
2
votes
1answer
36 views

Image of isometric immersion

Let $M$ a metric space with the following property: For all isometric immersion $f:M\to N$, the image of $f(M)$ is a open set in $N$. Prove that $M$ is empty set. A function $f:M\to N$, is called ...
1
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0answers
18 views

Range of values of $\frac {\epsilon}{\sup \delta}$ for the modified step function

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} ...
0
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0answers
14 views

Geers MPC metric - understanding magnitude error and phase error

It is regarding this question/subject Integrate two sets of Data and check similarity I'm applying the formulas mentioned [here]. But I still can't understand these error rates M and P. It is ...
2
votes
1answer
45 views

Prove the following theorem involving homeomorphic metric spaces.

I want to prove this theorem: If $(X,d)$ and $(X,d')$ be homeomorphic metric spaces, then they have the same convergence sequences. However, there exists homeomorphic metric spaces $(X,d), (X,d')$ ...
0
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1answer
26 views

Completion of metric-space, when “corresponding” Euclidean space is complete

I somehow couldn't find the answers to several probably simple questions. I am new to the topic, thus, please excuse any lack of knowledge. Let $A_1\subseteq\mathbb{R}^n$, such that the metric-space ...