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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Supposed X and Y are metric spaces and $f:X \rightarrow Y$ and $g:X \rightarrow Y$ are continuous prove $\{x \in X:f(x)=g(x)\} $ is a closed set

Supposed X and Y are metric spaces and $f:X \rightarrow Y$ and $g:X \rightarrow Y$ are continuous prove $\{x \in X:f(x)=g(x)\} $ is a closed set I don't have any idea where to start. Any suggestions? ...
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66 views

Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
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On the definition of Jordan curves

I read that the definition of Jordan curve is that it is homeomorphic to $S^1$. Is this equivalent to say that the curve is closable, continuous and non-self-intersecting? I'm not sure if closable is ...
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38 views

Show the following is Cauchy:

I am trying to prove that the Euclidean Norm/inner product on C([0,1]) does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in ...
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56 views

Prove that the union of the interior of a set and the boundary of the set is the closure of the set

I'll denote closure of A with $A_C$ because I cant get the bar for some reason. also $Int(A)$ is interior of A, $Bdry(A)$ is the boundary of A and A' the accumulation points. I'm trying to prove the ...
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36 views

metric spaces, $B_r(x)=B_s(y)$, is $y=x$ and $r=s$?

Let if $B_r(x)$=$B_s(y)$ for some $x$,$y$ in metric space $M$ and $r$,$s$ $\in$ $R$. Is true $x=y$? Is true $r=s$?
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75 views

Show the usual metric on $C([0,1])$ does not give rise to a complete metric space.

I know that I need to show there is a Cauchy sequence which converges to a point outwith $C[0,1]$ to show this. Is this the right way of going about it (I can only think of $f_n (x) = x_n = ...
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44 views

(i) Show that T is continuous on $(X,d)$. (ii) Show that T is continuous on $(X,d_{2})$.

Let $K(t,s)$ be a continuous function on $[0,1]\times{[0,1]}$. Let $X=C[0,1]$ be the set of continuous functions defined on the interval $[0,1]$. Define the mapping $T:X\rightarrow{X}$ by: for every ...
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98 views

$\ell^{\infty}(\mathbb N)$ is not a separable space

I have to prove that $\ell^{\infty}(\mathbb N)$ is not separable. My attempt Consider a SUBSET $V$ of $\ell^{\infty}(\mathbb N)$ consisting of bounded sequences that have only $0$, $1$ entries, e.g. ...
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31 views

Separability in Metric Spaces

Let $(X,d)$ be a non-separable metric space. My question is the following: does there exist some $\epsilon > 0$ and some uncountable subset $S$ of $X$ such that $d(x,y) > \epsilon $ for any $x, ...
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15 views

Finding points on both sides of a point of a perfect subset of R

I have a perfect set S intersected with the interval $ [a,b] $ and know that the intersection is perfect and non-empty in $ \mathbb{R} $. Given a point x, I construct a sequnce $ y_n $ such that $ y_i ...
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74 views

Closure of an open ball equal to the closed ball

If $X$ is a discrete space (metric). Then the closure of a open ball $B_1(x)=\{x\}$ is $B_1(x)=\{x\}$, and the closed ball is $X$, therefore do not coincide. You know another example such that: ...
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40 views

How to prove that for any $n\in N$, there exists a subset of real line that has nonempty $(n-1)^{th}$ derived set but an empty $n^{th}$ derived set?

How can we prove that for any positive integer n, i.e., $n\in \mathbb{N}$, there exists a subset of real numbers, i.e., $E\subset \mathbb{R}$, that has nonempty $(n-1)^{th}$ derived set but an empty ...
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$p$ is in $\operatorname{closure}(S)$ $\Leftrightarrow$ any ball centred at $p$ contains some point of $S$

I want to prove that $p$ is in $\mathrm{closure}(S)$ if and only if any ball centered at $p$ contains some point(s) of $S$. Where S is some subset of underlying set E of metric space (E,d). The ...
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77 views

Show this metric generates the product topology on $X$

Let $(X_n, d_n)$ be a sequence of metric spaces. Show that the function $ d: X \times X \to \mathbb R^+$ on the product space $X: = \prod_n X_n$ defined by $$d ((x_n)_{n = 1}^\infty, ...
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29 views

How to show in general a certain type of metric generates a certain type of topology?

If we're given a set $X$ together with a metric $d: X \times X \to \mathbb R^+$, and a topology $ \mathcal F \subset 2^X$. What do we need to do in order to show this metric generates this topology?
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32 views

triangle inequality on a given metric

$X$ be set consisting of all sequences $(x_1,x_2, \dots)$ s.t $x_i \in \mathbb R$ and $\sum x_i^2$ converges I need to prove triangle inequality for the metric on $X$ given by, $d(x,y) = [ ...
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64 views

Conditions to make a function a metric on $\mathbb{R}$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions ensure that $d(x,y)=|f(x)-f(y)|$ defines a metric on $\mathbb{R}$ Let $g:[0,\infty) \to \mathbb{R}$. What conditions on $g$ ensure that ...
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Notation Question: What does $B(0,1)$ mean when it comes to metric spaces?

I have to draw the unit balls $B(0,1)$ in $\mathbb{R}^2$ with respect to several metrics, however I am not certain whether this means a unit ball centered at $(0,1)$ or at $(0,0)$? Thanks for your ...
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31 views

Show these sequences converge and determine the limit of each.

Using the definition of convergence in metric spaces, show that the following sequences converge and find its limit. 1.) $a_n(x)=\frac{n}{n+1}x^2+\frac{2}{n}x+3$ in $(C[0,1],||.||_1)$ To begin we ...
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43 views

Metric completion of universal covering of punctured plane

It is known that the universal covering of the punctured plane $\mathbb C\setminus\{0\}$ is $\exp:\mathbb C\to\mathbb C\setminus\{0\}$. In real coordinates, $f=\exp:\tilde M=\mathbb R^2\to M=\mathbb ...
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How can we show whether this is a metric or not? [closed]

Define a function $d(x,y)=\arctan|x-y|$ for $x,y\in\mathbb{R}$. Is this a metric on $\mathbb{R}$?
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Metric definition example

Let $(X_1,d_1)$ and $(X_2,d_2)$ be two metric spaces. Define a new metric space $X=X_1 \times X_2$, such that for $x=(x_1,x_2)$, $y=(y_1,y_2)$, we have $$d(x,y)=\sqrt{d_1(x_1,x_2)^2+d_2(y_1,y_2)^2}$$ ...
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1answer
65 views

Every metric space has a $(1, 1)$-net

I'm trying to show that every metric space $X$ has a $(1, 1)$-net but struggling - surely $(1, 1)$ is just arbitrary and I've run out of obvious subgroups of $X$ to play with. Any help plz! Here a ...
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1answer
60 views

Where does the power $2$ come from in the Pythgorean theorem?

So $$a^2+ b ^2 =c^2$$ in a right triangle, but where does the power $2$ come from? I know we can use different metrics in the Euclidean space. If we use the $p$-metrics, where $p$ is in place of ...
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33 views

show that $\lim_{x \to a} f(x)= b$ is equivalent to $f(x_j) \to b $ as $j \to \infty$ for any $(x_j) \subset U / \{a\}:x_j \to a$

Could someone please tell me if $$\lim_{x \to a} f(x)= b$$ is equivalent to $f(x_j) \to b $ as $j \to \infty$ for any $(x_j) \subset U / \{a\}:x_j \to a$ If it is, how would I show that it is? ...
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64 views

Estimating the number of connected components of a curve contained in a given set

Let $X$ be a metric space and $\Omega\subset X$ an open set. Take $x\in\Omega$ and choose $r>0$ such that the open ball $B(x,r)\subset B(x,2r)\subset \Omega$. Let $\gamma:[0,1]\to X$ be a Lipschitz ...
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26 views

Number of connected components of a given curve inside a particular set

Assume that $(X,d)$ is a metric space and let $\Omega\subset X$ be a open set with $\operatorname{int}(\overline{\Omega})=\Omega$. Let $\alpha :[0,1]\to X$ be a Lipschitz curve and consider the two ...
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Topological interpretation of the following equivalence.

We assume $\{X_n\}_{n\in\mathbb{N}}$ and $X$ are random variables from $\{\Omega,\mathcal{F},\mathbb{P}\}$ to $(S,d_s)$, wehre $S$ a separable metric space. One can establish the following ...
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30 views

Is this claim harder to prove for arbitrary metric spaces than for the reals?

Let $X$ and $Y$ be metric spaces. Define the distance between functions $f, g$ from $X$ to $Y$ as $$d(f, g) = \sup_{x \in X} \frac{d(f(x), g(x))}{1+d(f(x), g(x))}$$ Is it true that if $f_n:X \to Y$ ...
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58 views

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
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44 views

embedding discrete metric into manifold?

True or false: "Any edge-weighted undirected graph can be isometrically embedded into some Riemannian manifold". "isometric embedding" here means that for any pair of nodes, their shortest path ...
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Using the hypothesis that an equicontinuous sequence is pointwise bounded to show it's uniformly bounded

I'm making steady progress in this problem, but it is a little unclear on where exactly to use the fact that $\{f_n\}$ is pointwise bounded. I was going the route of a compactness argument, but I'm ...
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1answer
46 views

A set is open in two metric spaces?

For $(X,d_1)$ and $(X_2,d_2)$ is two metric space, set $(X=X_1 \times X_2)$ and $d(x,y)=max\{d_1(x_1,y_1),d_2(x_2,y_2)\}$ , $\overline{d}=d_1(x_1,y_1)+d_2(x_2,y_2)$ with $x=(x_1,x_2),y=(y_1,y_2)\in ...
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Showing the unit ball in $l^{\infty}$ is not compact. [duplicate]

If $l^{\infty}$ is the set of bounded sequences of real numbers with norm $||x||_{\infty}$. To do this I have tried to use the fact that a metric space is compact iff it is sequentially compact. So ...
2
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71 views

Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
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36 views

Set that is closed and bonded, but not compact?

Let $\mathbb Q $ be the set of rational number with d(p,q) = |p-q| and E be the set of all p $\in \mathbb Q$ such that $2 < p^2 < 3$. Intutively, I think about closedness using subspace ...
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Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric space.

Show $\rho (x,y)=\dfrac{d(x,y)}{1+d(x,y)}$ is a metric on the metric space $X$, equipped with the Euclidean metric $d$. I've already shown that the positivity $\rho(x,y)\geq 0$, the symmetry ...
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44 views

Show that $\ c_X(p,q) \le d_X(p,q)$, for $ p, q \in X$

Update I'm trying to show the Corollary, but I have stuck...That is: For any complex space $X$, we have: $$\begin{align} (1).\ c_X(p,q) &\le d_X(p,q),\ \text{for}\ p, q \in X \\ (2).\ ...
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63 views

Construct such $d$ that $(\mathbb{R} \setminus \mathbb{Z}, d)$ is complete metric space

Good evening! I had topology exam yesterday and I had a question that gave me problems. Lets consider $\mathrm{G} = \mathbb{R} \setminus \mathbb{Z} $ , i.e. real line without integrals, where $\rho$ ...
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30 views

Show that $d_g$ is a metric on $l^1$.

On the space $l^1$ of complex valued sequences $(x_n)$ such that $\sum|x_n|<\infty$, define for $x=(x_1,x_2,\cdots)$, $y=(y_1,y_2,\cdots)$ the metric $d_f$ by ...
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32 views

Hausdorff metric and convex hull

Let $X$ be a normed linear space and $A, B \in P(X)$. We define $\overline{co}(A) =$ the closure of the convex hull of $A$. Let $h$ denote the usual Hausdorff metric. We need to show that: $$h( ...
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90 views

Show that $d(x,y)$ in a metric on $X$.

Let $d_a(x,y)=7|x-y|$ and $d_b(x,y)=|x+y|$ be metrics on set $X$. Show that $d(x,y)=d_a(x,y) + d_b(x,y)$ is also a metric on $X$. Would I be correct in writing $d_a(x,y) + d_b(x,y)$ as $7|x-y| + ...
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What does metrizablity means

Let $(X,T)$ be a topological space, and let $f$ be a homeomorphism form $(X,T)$ to any metric space. Than we say that $(X,T)$ is metrizabile. This means that X can be equipped with a metric and can be ...
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1answer
42 views

Closed sets and sequences in Metric spaces

Suppose $x \subset X$ is a closed set, the sequence {$ {x_j}$}${ } \subset F$ and $x \in X$. Show that if $x_j \to x$ as $j \to \infty$, then $x \in F$ Okay so I really don't know where to start with ...
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50 views

Metric spaces, compactness, how to solve this with the open covering property?

I have an excercise I am supposed to solve with the open covering property. I am not able to do that, but I can solve it with another method. I have two questions. 1. Is my method of solving it ...
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1answer
60 views

Metrizability under homeomorphism?

Is metrizability preserved under homeomorphism? That is, suppose that you have a topological space $(X, \tau_1)$ whose topology comes from a metric $d$, and you have another topological space $(Y, ...
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50 views

Sequence of $r_n:=\frac{x_{n+1}}{x_n}$ where $x_n$ is a sequence

Let $f:X\rightarrow X$ be a function on some complete metric space $(X,d)$. Let, $x_n$ is a sequence in the metric space defined by $x_{n+1}=f(x_n)$ and starting from $x_0$. My questions are (1) ...
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41 views

Is the convergence of a sequence independent of the chosen metric?

Given a metric $\rho$ on $X$ and a sequence $x_n$ in $X$. Does the convergence of $x_n\to x$ under $\rho$ also imply the convergence to the same limit under any other metric $\sigma$? I don't know th ...
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2answers
38 views

Problem about complete metric spaces: proving that the Banach fixed-point theorem requires the map to be a contraction map

Prove that there exists a nonempty complete metric space $(X,d)$, and a function $f:X \rightarrow X$, such that $d(f(x),f(y))<d(x,y)$ whenever $ x \neq y$, and such that $f(x) \neq x$ for all $x ...