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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why is any singleton set not open in the set of rational numbers

I know that this is true and is used to prove that $\mathbb{Q}$ is not a discrete metric space, but I can't figure out, why is it true ?
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54 views

Prove that $\mathbb{N}$ with the metric $d(m,n)=\lvert m^{-1} - n^{-1}\rvert$ is a discrete metric space

Prove that $\mathbb{N}$, along with the metric $d(m,n)=\lvert m^{-1}-n^{-1}\rvert$, is a discrete metric space. I am stuck with this one, I don't know how to proceed ? Any help will be ...
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23 views

Sequentially compact $\Rightarrow \inf_{x\in A}\varepsilon_{x}=:2\varepsilon_{0}>0$

I've highlighted the sentence which I dont quite understand. Maybe I should say, that I interpret $\inf_{x\in A}\varepsilon_{x}$ as the greatest lower bound of $E_{x}$ - if this is wrong, then I need ...
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1answer
155 views

Prove that N with its usual metric inherited from R is a discrete metric space

I was trying to prove this result. I started out by taking some arbitrary subset, S of N,and finding its boundary points. Boundary points of S is the set of all points x of N whose distance from S ...
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2answers
41 views

Question on Compactness

Let the metric space be the real numbers with the usual distance formula. Let $E$ be an open interval from $1/8$ to $2$. Then $E$ would be compact if every open cover of $E$ has a finite cover. I know ...
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34 views

A metric is open in itself

For a metric space $X$, I want to find a neighbourhood of any point $p \in X$ which is a subset of $X$. How to find specify that $r$
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388 views

Totally bounded subset in complete metric space implies compact?

I am reading the book Elements of Functional analysis by Kolmogorov and Fomin. In chapter 2, section 16 on compact metric spaces the author poses the following theorem which he demonstrates ...
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0answers
95 views

Show that the distance between these two sets is not bounded.

I have a homework question that asks: "Consider the curve $\gamma : [1, \infty] \to \mathbb{R}^2$ defined by $\gamma (t) = \langle t \cos (\ln t), t \sin (\ln t) \rangle$. Show that this curve is ...
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103 views

Proof of an application of the contraction mapping theorem to differential equations

Please consider the theorem below together with the first part of its proof. 1) Why is M closed? 2) Why is M complete? 3) Why is the final integral a continuous function? (The curvy C denotes the ...
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62 views

Proving that half an isometry is a homeomorphism

Let $(K,d)$ be a compact metric space and $f:K\rightarrow K$ such that $$\forall x \in K, \forall y \in K, d(f(x),f(y)) \geq d(x,y)$$ Prove that $f$ is a homeomorphism. What I managed to prove is ...
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179 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
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43 views

what is the meaning of the “closure of a sequence ”

Suppose $X$ is a metric space, $z$ is in $X$ and $(x_n)$ is a sequence in $X$. Then what does it mean to say that, $z$ is in the "closure of every tail of $(x_n)$." What does "closure" of every ...
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1answer
39 views

Convergence criteria for interior

My book gives three equivalent statements as a theorem, under the heading "Convergence criteria for interior" : Suppose X is a metric space, z is in X and S is a non empty subset of X. First,z is in ...
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3answers
72 views

relation between metric and topology

I have two questions regarding the relationship between metrics and the topology that they generate : First , if the metric changes then, is it necessary that the topology would also change ? ...
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1answer
42 views

exception for disproving this sufficient condition

The following is sufficient but not necessary condition for topological equivalence: for each x $\in$ X, there exist positive constants $\alpha$ and $\beta$ such that, for every point y $\in$ X ...
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3answers
93 views

Uniqueness of Limits of sequences

I was reading about limits of sequences recently, and I came across this fact that, "there are situations in which sequences which may converge to more than one point. Is that really possible ? Till ...
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1answer
39 views

Is there a generalized metric, with these following properties?

I have come to know from Wikipedia article about what are called generalized metrics, and that they differ from the regular metric definition in terms of the properties/requirements they have to ...
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923 views

Why is it that in a discrete metric space only eventually constant sequences are convergent?

I just read this result and was wondering what is the intuitive idea behind this ?
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95 views

Compactness of a set

Let $A = \{1/n\mid n\in\Bbb N\}\subset\Bbb R.$ I understand that a set must be closed to be compact, and every open set is not compact. I know that $A$ is not closed because $0$ is a limit point, but ...
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345 views

Bounded derivative implies bounded function?

By the following theorem, it suffices to show that $\{F_n: n\in\mathbb N\}$ is equicontinuous and bounded: If $f_k$ is a sequence in an equicontinuous and pointwise bounded set of maps from a ...
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1answer
40 views

Prove that for two vectors x,y over GF(q), the number of vectors that are closer to x is the same as the number of vectors that closer to y.

Let $x,y\in\mathbb F_q^n$ be vectors. We'll define: $X= \{ u\in\mathbb F_q^n \mid d(x,u)<d(y,u)\}$ $Y= \{ u\in\mathbb F_q^n \mid d(y,u)<d(x,u)\}$ Prove that $|X|=|Y|$. Well. ...
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22 views

What is contained in $E_{x}$?

I can't exactly understand what is contained in $E_x$. As I think of it, $E_x$ contains distinct points $r$, for which the below holds. But since the author states that the interval $(0,r)$ is ...
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1answer
184 views

Is a set of bounded functions bounded?

Please consider the following question (note that $C_b$ is the space of bounded continuous functions): Let $f_k$ be a convergent sequence in $\mathscr C_b(A, \mathbb R^m)$. Prove $\{f_k \mid k = ...
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1answer
798 views

Convergence of a constant sequence and an eventually constant sequence

I have to prove that a constant sequence and an eventually cinstant sequence is always convergent. I tried to do this as follows : I considered (xn) in X to be a constant sequence such that (xn) = ...
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186 views

Metric induced by a norm - what conditions should this metric meet?

$(I)$ I've been browsing some problems concerning metrics not induced by norms, and I've found a comment that said that such a metric should be a concave monotone function. Here is the post I'm ...
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43 views

Trying to show that the interior of a given set is contained in another given set

How do I show that in the space of bounded continuous functions from $\mathbb R$ to $\mathbb R$, if there exists an open $\epsilon$-ball about $g$ that is contained in the set of functions that ...
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2answers
104 views

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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1answer
93 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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1answer
107 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
311 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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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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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
132 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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1answer
34 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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96 views

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 ...
4
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1answer
555 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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1answer
73 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
100 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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1answer
673 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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2answers
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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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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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1answer
174 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 $x\neq 0_{\mathbb{K}}$; $\left|0\right|=0$ ...
3
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1answer
184 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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485 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 + ...