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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Boundary of a ball

Show that the open ball $B(0,1) = \{(x,y): x^2 +y^2 < 1\}$ has the boundary $x^2+y^2=1$. I understand that the boundary is the closure of the ball minus the interior. So, if i can show that the ...
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81 views

Interior points in topology

Let $X= \mathbb{R^2}$ with subway metric. Here subway metric is the Paris metric. Let $A= [-1,1] \times \{0\}$. What is the interior point of $A$? I would say it is $(-1,1) \times \{0\}$ but I got it ...
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95 views

Convergence of the sequence, $\frac 1n$

Why does the sequence $\frac 1n$, where $n$ is a natural number , does not converge when R is endowed with the discrete metric ?
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94 views

Criteria for convergence of a sequence in a metric space

Let $X$ be a metric space, $z$ be in $X$ and let $(x_n)$ be a sequence in $X$ Using the fact that, Every open subset of $X$ that contains $z$ includes a tail of $(x_n)$, I have to prove that ...
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109 views

boundary points of an infinite subset of a metric space

Does any infinite subset of a metric space have boundary points ? I know that the set of boundary points of a metric space is empty.But i am not very sure about whether, this is true for any ...
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134 views

Compactness and closed balls

Let $E$ be a compact metric space, such that $\{U_i\}_{I\in I}$ is a collection of open sets whose union is $E.$ Show that there exists $\epsilon>0$ such that any closed ball in $E$ of radius ...
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1answer
327 views

Clarification on this corollary of the Arzela-Ascoli Theorem

I am given the following corollary without proof: A family of continuous functions on a compact metric space into $\mathbb R^m$ is compact iff it is closed, equicontinuous and bounded. Does ...
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1answer
67 views

Counterexample of Metric Spaces

I know the result that if X and Y are 2 metric spaces with Y complete and f is uniformly continuous on a dense subset D of X then f can be continuously extended to X. Can someone show that this ...
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1answer
68 views

Metric Space and ordered field

If we have an ordered field $ \mathbb{F} $, can we consider a natural metric involved with this space? What should be this metric? thanks
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1answer
48 views

Metric in an ordered field

Suppose that we have $ \mathbb{F} $ an ordered field with a metric d and $x,y \in \mathbb{F} $ non negative numbers. It is possible to affirm that if $ x \leq y $ then $d (x,0) \leq d(y,0) $? If we ...
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1answer
135 views

Question on a corollary of the Arzela-Ascoli theorem

I am given a corollary of the Arzela-Ascoli theorem, and I've substantially rephrased it to this: If $S$ is an equicontinuous and pointwise bounded set of functions with domain a compact metric ...
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35 views

Distance between differential operators

Given two differential operators say $D_1$ and $D_2$ is there any meaningful way to define distance between them, does there exist some metric $d(D_1,D_2)$ that satisfies all the necessary properties? ...
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If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$

If $(X,d)$ is a complete metric space and $A$ is closed in $(X,d)$ then show that for each $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0).$ I tried this problem several ...
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560 views

closed,bounded not compact

Hi I was asked to prove that: if $S =\{ x \in \Bbb R : d(x,0) = 1 \}$ then $S$ is a closed and bounded set. The set $S$ contains only two points: $-1,1$,(it should not be a problem to prove that is ...
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74 views

Closure criterion for the convergence of sequences in a metric space

Suppose $X$ is a metric space, $z\in X$, and $(x_n)$ is a sequence in $X$. Then according to the closure criterion for convergence of $(x_n)$ in $X$ we have that, $$\{z\} = ...
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2answers
101 views

What does it mean for a set to be compact in another set?

I am given the following definition: Let $B$ be a set of continuous maps with domain a metric space $A$ and codomain a metric space $N$, and $B_x=\{f(x):f\in B\}$. $B$ is pointwise compact ...
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690 views

Separable metric space has a countable base

A collection $\{V_{\alpha}\}$ of open subsets of $X$ is said to be a $\textit{base}$ for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have ...
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Is it possible for $b[x;r) = b[y;s)$ when $x \neq y$ and $r \neq s$?

I know it is possible, for instance if we consider a non empty set $X$ with the discrete metric, then for each $x \in X$ the balls $b[x;r)$ for $r \in (0,1]$ are equal to the singleton set $\{x\}$. ...
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99 views

Can a metric be recovered from the collection of open balls it produces?

My book says that this cannot be done unless the radius and the centre of the ball are known. I don't understand, why is it important to know the radius and the centre of the ball ?
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68 views

Separability and open balls

I have a super basic question that for some reason has been eluding me for quite a while. This question actually came up in the context of weak convergence of probability measures on the space ...
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181 views

Symmetry of a Manhattan Distance

I am having trouble with proving that the Manhattan distance (also known as Taxicab geometry) is a metric by satisfying the condition of symmetry. Can anyone point me in the right direction?
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50 views

Absolute value defined in a field

Let $\mathbb{K}$ be any field. Let $\left|\cdot\right|:\mathbb{K}\longrightarrow\mathbb{R}$ be a function which satisfies $\left|x\right|>0$ if $x\neq 0_{\mathbb{K}}$; $\left|0\right|=0$ ...
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188 views

If $X$ has a metric $d$ then the topology induced by $d$ is the smallest topology relative to which $d$ is continuous

(Munkres, p. 126, Ex. 3) Prove the following: Let $X$ be a metric space with a metric $d$. Let $X'$ be a topological space that has the same underlying set as $X$; i.e., $X' =X$ but $X'$ might have ...
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501 views

proving Cartesian product of two metric space is a metric space

I saw somewhere that Cartesian product $X = X_1 \times X_2$ of two metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ can be made into a metric space $(X,d)$ like following: $d(x,y) = (d_1(x_1,y_1)^p + ...
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Suppose $X$ and $Y$ be metric space, $A \subseteq X$, $f$ be a continuous map from $A$ to $Y$

Suppose $X$ and $Y$ be metric space, $A \subseteq X$, $f$ be a continuous map from $A$ to $Y$, then: I, $A$ is compact, then $f(A)$ is compact; II, If $A$ is bounded, then $f(A)$ is uniformly ...
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Is there an analogue/primitive of PCA which can be in a metric space rather than a vector space?

Principle component analysis PCA is done in a vector space, basically projecting a given vector onto the eigen vectors of the covariance matrix. I'd like to think of a primitive analogoue of PCA, ...
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572 views

Equivalence to properly discontinuous action

Let $X$ be a metric space and let $G$ be a group of homeomorphisms $X \to X$ acting on $X$. We say $G$'s action is properly discontinuous in case for every $x \in X$ and compact $K \subseteq X$, there ...
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1answer
155 views

How to project 3D plots on a 2D coordinate system without losing the metric scale?

I've been collection data of a river bank last week and I need to plot the cross sections of the data. The issue is, that the data taken consists of 3 coordinates: easting, northing and elevation. ...
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1k views

Closed set as a countable intersection of open sets

Let's take a metric space. Then any closed set can be written as a countable intersection of open sets. How can I prove that?
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34 views

Estimating/Reconstructing the distance matrix by given pairwise distance of a subset of points

Given a set of points $X$ which separated into two subsets $X_1$ and $X_2$ i.e. $X_1 \cup X_2 = X$ and $X_1 \cap X_2 = \emptyset$ We have the pairwise distance matrix $M^1$,$M^2$ of set $X_1$, $X_2$ ...
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Does $\mathcal{L}^2(\mathbb{R})$ form a metric space with this distance/similarity measure?

Consider the set $\mathcal{L}^2(\mathbb{R})$, where two functions $f$ and $g$ are said to be equal, if they agree almost everywhere. I would like to define a distance/similarity measure and would like ...
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lebesgue measure is metric outer measure

This question is driving me crazy. I need to prove that Lebesgue measure is metric outer measure. Unfortunately, I get lost. All I have is because $m$ is Lebesgue measure, $m^*(A \cup B) < ...
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618 views

Not Complete Metric Space

Suppose I have a metric $\rho: R^n$ x $R^n \rightarrow R$ defined by $\rho (x, y) = \frac{d(x, y)}{1+d(x, y)}$ and I need to show that $(R^n, \rho)$ is not a complete metric space? Can anyone suggest ...
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638 views

Is every normed vector space a metric space?

I was trying to prove that every normed space is a metric space, and the first three proprierties came natural. However, when faced with proving the triangle inequality I had a bit of problems. I ...
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44 views

Prove $(X,d)$ is a metric. Where d is the number of entries a vector is not equal.

I am reading through my applied analysis book and trying to prove all of the things that have not been proven in order to prepare for an exam. Here is one of the examples: Let $X$ be the set of ...
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69 views

Was this metric induced by a norm?

Is the metric $$d(u,v)=\frac{||u-v||_v}{1+||u-v||_v}$$ induced by a norm? My attempt at an answer: Suppose that it was then, there would be a norm $||.||_m$ such that ...
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Completeness of uniform metric

Let $C[0,1]$ be the set of continuous real valued functions on $C[0,1]$. Show that $(C[0,1],\rho_\infty)$ is complete. Is $(C[0,1],\rho_1)$ complete? Justify your answer. Here, ...
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Special Metric on the set of intervals

Let $X=\{[a,b]\ a,b \in R$ and $a<b \}$ and Let $y=\{(a,b)\ a,b \in R $ and $ a<b \}$ then we define a metric on X as: $$d( [a,b], [c,d] )= \inf \{\epsilon >0 : [a,b]\subseteq [c-\epsilon, ...
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A question on Hamming distance [closed]

Let $X$ be the set of all ordered triples of zeros and ones. Show that $X$ consists of eight elements and a metric $d$ on $X$ is defined by $$d(x, y) = \text{number of places where}~~ x~~ \text{and}~~ ...
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1answer
146 views

topologically equivalent spaces

I was wondering if there exist a metric $d:X\times X\longrightarrow\mathbb{R}$ (with $X:=(-1,1)$) such that $(-1,1)$ is complete and that $(d,X)$, $(d_{usual},X)$ are topologically equivalent? I know ...
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Metric space statements

Let $(X,d)$ be a metric space and let $A, B$ and $E$ be subsets of $X$. The boundary of a subset $E$ of $X$, is denoted by $\partial E$, and is defined as $\partial E= \overline{E} \cap ...
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100 views

Let $(X,d)$ be a metric space and let $A \subset X$.

Let $(X,d)$ be a metric space and let $A \subset X$. Prove that: (a): $X-int(A)=\overline{X-A}$ (b): $X-\overline{A}=int(X-A)$ (Notation: $\overline{E}$ is the closure of $E$ where ...
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Show that $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed

I have $\mathbb{R}$ with the euclidian metric $|x-y|$ for $x,y\in \mathbb{R}$. I want to show with arguments that the set $D =\{ x + y \mid x \in (0,1) ,y \in [1,2) \}$ is open or closed. As a ...
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Determine if the set $C =\{ \frac{1}{n}|n \in \mathbb{N} \}$ is open or closed in the euclidian metric

I have $\mathbb{R}$ with the euclidian metric$\sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}$. I want to show with arguments or a proof that the set $C =\{ \frac{1}{n}|n \in \mathbb{N} \}$ is open or closed. I ...
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83 views

Topology induced bycone metric

Is cone metric define atopology as same as the topology define by ametric? I have tried to prove it by theorems that joined them
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Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
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81 views

Is a finite set in R² open?

How can a finite set E in R² not be open? Isn't every point of E an interior point of E (since the point will be contained in its own neighbourhood)? And if every point in E is an interior point, then ...
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neighbourhood of x in a metric space

Does the point $x$ belong to the $\epsilon$-neighbourhood of $x$? According to the definition, the neighbourhood of $x$ consists of all $y$ such that $d(y,x)< \epsilon$. Does $x$ belong to the ...
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46 views

Only constant curves rectifiable

I would need a hint on the following standard exercise. Let $0 < a < 1$ and $(X,d)$ a metric space. Let $d^a$ be the corresponding snowflake metric $d^a(x,y) = (d(x,y))^a$. Show that the only ...
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Is this set open in the Euclidean topology on the plane?

Let $\mathbf{R}^2$ be the two-dimensional Euclidean space, and let $$ A := \{ (x,y) \in \mathbf{R}^2 | \, \, \, |x| < \frac{1}{y^2+1} \}.$$ Then how can we establish (preferably using the ...