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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The distance between an element and a subset of a metric space.

I got stuck on an assignment. Can you help me to solve this? Let $(X,d)$ be a metric space, and let $C$ be a subst. Define the function: $$ f \quad : \quad X \longrightarrow \mathbb{R} \quad : ...
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Metric Space Properties

I am revising my notes on metric spaces and one of the Theorems is stated but the proof has been ommitted and was looking for some help as I don't know where to start. Let $(X,d)$ be a metric space, ...
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How can the y-axis in $\mathbb{R^2}$ be open?

I have read that $\{(x, \frac{1}{x}): x \neq 0\}$ is closed in $\mathbb{R^2}$. So hence the complement of this set, $\{x = 0\}$, i.e. the y-axis must be open? But we cannot put an open ball with ...
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111 views

Completeness and separability of $B(X)$ (bounded linear operators on $X$)

Assume $X$ is a separable Banach space with norm $|| \cdot||$. Consider $\{f_n\}_{n \in N}$ a countable dense subset of $X$ and equip $B(X)$ (bounded linear operators on $X$) with the following ...
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100 views

In what metric spaces is a closed and bounded set compact?

Is there a characterization of a metric space $X$ such that for every $A\subseteq X$, $A$ is compact iff $A$ is closed and bounded? Something that generalizes $\mathbb R^n$?
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454 views

Definition of a nowhere dense set

I'm currently studying metric spaces through Gamelin and Greene's Introduction to Topology. While studying about completeness I got stuck with this concept of nowhere dense subset. The book defines a ...
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52 views

In a metric space, if $A$ is open and $B$ is closed, is $A + B$ open or closed?

Let $A, B \in E^n$, and consider their sum $A + B = \{x+y \mid x \in A, y \in B\}$. Suppose that $A$ is open and $B$ is closed. Is it always true that $A+B$ is open? Is it always true that $A+B$ is ...
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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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Is $\mathbb R^{\omega}$ homeomorphic to $\mathbb R^{\omega} \times \mathbb R^{\omega}$?

As a study exercise, I'm trying to find a topological space $X$ which is homeomorphic to $X \times X$. I began thinking of simple examples involving $\mathbb R$ but then realized my best bet would be ...
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53 views

Is a finite subset of a complete metric space again a complete metric space?

The space $(\mathbb{R}^2, d)$ where $d(x,y)=max \{|x_1-x_2|, |y_1, y_2|\}$ for $x=(x_1, y_1)$ and $y=(x_2, y_2)$ $\in X$ is a complete metric space. Let $X=\{(0,0), (-\frac{1}{8}, 0), (0, ...
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315 views

What minimizes the Chebyshev Distance?

For an arbitrary number of dimensions, I know that the mean minimizes the distance using the L2 norm and that the geometric median minimizes the distance function using the L1 norm (though I have yet ...
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89 views

Is this a metric on R?

For $x,y \in \mathbb{R}$, define $d(x,y) = \sqrt{|x-y|}$. Is this a metric on $\mathbb{R}$? It's clear that $d(x,x) = 0$ and $d(x,y) = d(y,x)$ for all $x,y \in \mathbb{R}$. The triangle inequality ...
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42 views

prove the set of all spheres with rational center and radius is countable

Prove the set of all spheres in $\mathbb{R}^3$ with rational center and radius is countable. I have two ideas. Is either one better than the other? 1) let $(x,r)$ represent a sphere in ...
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45 views

Continuity of identity in $p$-adic $\mathbb Z$

Say we have the $p$-adic metric in $\mathbb Z$ defined as $$ d_p(a,b)= \left\{\begin{align} &0 & a=b \\ &p^{-r} : p^r\mid (a-b), p^{r+1}\nmid (a-b) & a\neq b \end{align}\right. $$ I'd ...
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70 views

Completeness & Closedness in Metric Spaces

If every (proper) closed subset of a metric space is complete, then is the whole space necessarily complete as well?
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What is this space with infinitely many different points with distance $1$ between any two different points?

I'm reading Mac Lane's: Mathematics, Form and Function: [...] There are also bizarre examples - such as "a space" with infinitely many different points, with distance $1$ between any two different ...
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218 views

Proving that a set of metric space is dense in $A$ iff there exists a sequence converging to $x\in A$

I'm using the following definition: A set $M$ of a metric space $(\frak M,\rho)$ is called dense in a set $A\subset\frak M$ if $$\forall \varepsilon>0,x\in A\exists y\in ...
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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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1answer
249 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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846 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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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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213 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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47 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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315 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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91 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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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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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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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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37 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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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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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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79 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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1answer
98 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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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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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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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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35 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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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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1answer
333 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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74 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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85 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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40 views

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 ...
3
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1answer
250 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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1answer
81 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 ...