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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Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
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42 views

Show that $d_V$ is a metric

Problem: For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $\mathbb{R}^2$ define: $d_V(p,q) = \begin{cases}1 & p_1\neq q_1 \ or\ |p_2 - q_2|\geq 1 \\ |p_2 - q_2| & p_1= q_1 \ and\ |p_2 ...
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1answer
33 views

How to show that this set is closed in $\mathbb{R}^n$?

For an open set $\Omega\subseteq\mathbb{R}^n$, let $K_j$ be the set of points $x$ of $\Omega$ such that $\text{dist}(x,\partial\Omega)\geq1/j$ and $|x|\leq j$. Question : Why is $K_j$ closed ? ...
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Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
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Statiscal Distance Properties

Please anyone could give me any idea of how prove the following property of statistical distance: d(AB,CD)=d(A,C)+d(B,D) Remenber that: $(X,d)$---> Metric ...
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4answers
40 views

Proving $d_1$ is a metric.

If $d$ is a metric on a set X, then $d_1 = \frac{d(x,y)}{1+d(x,y)}$ is also a metric. I have proved the other conditions of being a metric except the triangle inequality. Please help!!
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1answer
19 views

Sequential continuity on metric spaces

Please give me a hint for proving this statement: Let $(X,d)$ and $(Y,d')$ be metric spaces, $f$ a function from $X$ to $Y$. If $f^{-1}(B) $ is closed in $X$ for all closed subset $B$ of $Y$, then ...
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20 views

Problems in metric space including matrices. [on hold]

Let $M(n, \Bbb R)$ denote the set of a real $n \times n$ matrices. We can always define a linear isomorphism between $M(n, \Bbb R)$ and $\Bbb R^{n^2}$....where the isomorphism is defined as for any ...
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1answer
31 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
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Exercise on Metric space

I hve this exercise it is very simple but i don't know how to write the answer Let $A$ be a nonempty set in $(E,d)$, for $\varepsilon>0$ we note $$V_{\varepsilon}(A)=\{x\in E, ...
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1answer
34 views

Separability of the Set of Bounded Functions over [0,1]

I'm working through Neal Carothers' Real Analysis and I'm stuck on trying to show that the set $B$ of bounded, real-valued functions over $[0,1]$ is not separable. The metric of this set is ...
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1answer
42 views

In a metric space we have $B(x,r) = B(y, s)$, is it necessary that $x = y$ and $r = s$??

If in a metric space we have $B(x,r) = B(y, s)$, is it necessary that $x = y$ and $r = s$? I think that the center of the balls i.e. $x$ and $y$ must be same but the radius $r$ and $s$ may not be ...
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0answers
11 views

How to prove a constructed set is a (n,ϵ)-spanning set for a [0,1] -> [0,1] homeomorphism

More specifically, I'm trying to figure out how to show that the following set is an $(n,ϵ)$-spanning set: $S = \{f^{-i}\big(\frac{j}{N}\big) \big| i = 0,1...n-1, j=0,1,...N\}$ where $N$ is selected ...
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1answer
16 views

Is there a Heine cretierion of liminf of a function?

Lately i've been struggling with understanding the meaning of $\liminf_{x\to x_0}f(x)$ assuming $f:X\to\mathbb C$ for $X$ a metric space, or for that matter $f:\mathbb R\to \mathbb R$. Could you give ...
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0answers
22 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
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2answers
19 views

Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
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1answer
21 views

What is an example of a connected subset of $\mathbb{R}^2$ where the interior is not connected?

In $\mathbb{R}^2$ with the usual metric, could this be an open disk, e.g. $ \{(x,y) : x^2 + y^2 \le 2\}$? Thanks in advance!
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1answer
25 views

A sequence of unbounded functions cannot converge uniformly to a bounded function. [closed]

Show that a sequence of unbounded functions cannot converge uniformly to a bounded function. I tried doing the problem using contradiction but failed!! Help Needed!!
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1answer
26 views

Net convergence in metric spaces

This is a question about convergence of nets which I don't quite understand yet. In metric spaces convergence of sequences encodes the topology but suppose we want to study convergence of nets even ...
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1answer
40 views

Showing that every set in a metric space is open and closed

Define: $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ I have shown that $(\mathbb{N}, d)$ Is a metric space. I have also been asked to describe all open balls of radii $1/3, 2/3 $ and $3$ in this metric space. ...
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1answer
54 views

Is it possible for an uncountable subset $S$ of $\Bbb R$ to satisfy $∂S = S$?

Is it possible for an uncountable subset $S$ of $\Bbb R$ to satisfy $∂S = S$? Please shed some light to it. I am not getting any clue to it.
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1answer
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A metric space in which no non-empty countable subset has empty interior.

A Question: Find a metric space in which no non-empty countable subset has empty interior. My Answer: In a discrete metric space $X$, $int(S) = S$ for all $S ⊂ X$. And if $S$ is non-empty (and ...
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1answer
56 views

When does a homeomorphism preserve Cauchy Sequence?

Is there any necessary and sufficient condition under which a the inverse function in a homeomorphism will preserve Cauchy Sequence ?
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98 views

How to prove the set $\{\cos(n) \mid n \in \mathbb{N}\}$ is dense in $[-1,1]$ [duplicate]

Prove that the set $\{\cos(n) \mid n \in \mathbb{N}\}$ is dense in $[-1,1]$.
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1answer
21 views

A meagre set is always contained in an $F_σ$ set made from nowhere dense sets.

In this page I have found a beautiful result that a meagre set need not be an $F_σ$ set (countable union of closed sets), but is always contained in an $F_σ$ set made from nowhere dense sets. Also ...
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1answer
27 views

Metric triangle inequality $d_2(x,y):= \frac{d(x,y)}{d(x,y)+1}$

$(X,d)$ is a metric space. $x,y,z \in X$ Now I have to proof that $(X,d_2)$ is also a metric space. To show that $d_2(x,y)=0 \leftrightarrow x=y $ and $d_2(x,y) = d_2(y,x)$ are correct was quite ...
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29 views

Properties of a metric

Consider the set $X=C([0,1])$. Define the function as $$\rho(f,g)= \int_{0}^{1}{x|f(x)-g(x)|dx}$$ $f,g \in X$. a) Show that $\rho$ is a metric in X. b) If $\lambda \neq 0$, then $$\lambda ...
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1answer
18 views

How to show that boundary of unit ball is empty in ultrametric spaces

Let $(S,d)$ be an ultrametric space. According to wikipedia, any ball must have empty boundary. Why is this true? I am unable to prove this.
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Is $d(x,y)=(x-y)^2$ a valid metric in $\mathbb R$?

Is $d(x,y)=(x-y)^2$ a valid metric in $\mathbb R$? So obviously $d(x,y)=(x-y)^2\ge0$ for all $x,y \in \mathbb R$ and equality iff $x=y$, and is also symmetric $d(x,y)=d(y,x)$. But how do I check ...
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1answer
35 views

$A$ and $B$ are connected subsets in a metric space X. Prove at least one of $ A\cup B $ or $ A\cap B $ is connected.

$A$ and $B$ are connected subsets in a metric space X. Prove at least one of $ A\cup B $ or $ A\cap B $ is connected. I'm not sure where to start for this one. All I know about multiple connected ...
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1answer
82 views

Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$

Use the Contraction Mapping Principle to show that $x=\frac19\sin\left(3x\right) + \sqrt{x}$ has exactly one solution $x\geqslant\frac{8}{9}$. I have literally no idea if this is right, please ...
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1answer
31 views

Countable union of compact sets is compact?

Let $A_0$ be a compact set (closed and totally bounded in some metric space) and consider a sequence of sets $A_n=\{x:d(x,A_0)<1/n\}$. For each $n$, $A_0\subset B_n\subset A_n$ is compact. ...
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27 views

Two geometrical objects in same dimensional plane are homeomorphic.

What can be a good way to prove that two geometrical objects in same dimensional plane are homeomorphic?? For example....to show that a circle and a ellipse is homeomorphic in $\Bbb R^2$ and a ...
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46 views

Product of Two Metrizable Spaces

I am having trouble with a practice exam question: $$\text{Show that if $X$ and $Y$ are metrizable, then so is $X\times Y$}$$ What I have so far: Given metric spaces $(X,d_x)$ and $(Y,d_y)$, I know ...
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Question on Uniform Continuity

Is it generally true that all uniformly continuous bijections $f: X \to Y$, where $X$ and $Y$ are metric spaces, have uniformly continuous inverses? If not, then what would be a counterexample, and is ...
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1answer
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$X$ contains at least two points & at least one isolated point. Prove $X$ is not connected.

Can we take two sets $G_1 = (x_1)$, where $x_1$ is the isolated point, and $G_2 = B(x_2;\epsilon)-(x_2)$ where $x_2$ is a limit point and show that the set- connectedness conditions hold? Help would ...
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General Relativity perturbation

Could anyone explain to me what I have misunderstood/missed when trying to understand this paper on GR perturbation? The paper is http://arxiv.org/pdf/0704.0299v1.pdf In equation 25 for $R_{00}$, ...
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36 views

To show that $f$ is continuous.

Let $f : \Bbb R^n \to \Bbb R^n $ be defined as follows: $f(x) = x$ if $\|x\| \le 1$ and $\frac{x}{\|x\|^2} $ if $\|x\| \gt 1$. Show that $f$ is continuous..Please help to start with the proof. I ...
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1answer
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Metric $p := p(x,y)= \min(|x-y|, 1- |x-y|)$ $x,y \in [0,1)^2$. Prove metric space is compact.

Help! I know that $X$ is Compact if every sequence in $X$ has a subsequence converging to a point in $X$. Also we have that $X$ is a bounded infinite subset in the real numbers. I think it's quite ...
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1answer
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Metrics on $\mathbb R^n$, Counting continuous functions and Open sets

Given the set $\mathbb{R}^n$ with metric $d$. We define continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ by open sets -we say that function is continuous iff the pre-image of every open set ...
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1answer
34 views

To show that any linear map $T: R^n \to X $ is continuous.

Let $X$ be any (not necessarily finite dimensional) Normed Linear Space . Show that any linear map $T: R^n \to X $ is continuous. Given any $\epsilon > 0$, there exists a $ δ > 0$ such that ...
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2answers
38 views

dist($\bar A,\bar B$) = dist$(A,B)$

Suppose $(X, d)$ is a metric space and $A$ and $B$ are subsets of $X$. Show that dist($\bar A,\bar B$) = dist$(A,B)$. I have shown that dist($\bar A,\bar B$) $\le$ dist$(A,B)$ but am unable to prove ...
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To show that $∂C = C$ where $C$ denote the collection of constant functions in $F$.

Consider the set $F$ of functions from $[0 , 1]$ to $[0 , 1]$ with the metric $(f, g) → sup${$|f(x) − g(x)| x ∈ [0 , 1]$}. Let $C$ denote the collection of constant functions in $F$. Show that $∂C = ...
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To prove triangle inequality for $d : \mathbb C \times \mathbb C \to \mathbb R$ ; $d(x,y):=\frac {|x-y|} {\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ [duplicate]

Is the function $d : \mathbb C \times \mathbb C \to \mathbb R$ defined by $d(x,y):=\dfrac {|x-y|} {\sqrt{1+|x|^2}+\sqrt{1+|y|^2}}$ a metric ? I can easily prove it is symmetric and positive-definite ; ...
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closure of the unit ball [duplicate]

Is the closure of the unit ball of $C^1[0,1]$ in $C[0,1]$ compact? For this let us take a sequence $x_n$ in $C^1[0,1]$ to show it has a convergent subseqence How to proceed with this.I am not so ...
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43 views

If $f$ is continuous & $\lim_{|x|\to {\infty}}f(x)=0$ then $f$ is uniformly continuous

Let, $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim_{|x|\to {\infty}}f(x)=0.$ Then prove that $f$ is uniformly continuous. I tried through the formal definition of uniform ...
2
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2answers
54 views

Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
2
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2answers
37 views

Proof that a certain function is uniformly continuous?

Consider a metric space $(M, {\rm d})$ and $y$ fixed in $M$. I want to prove that the function $f$ defined by $f(x)\colon={\rm d}(x,y)$ is uniformly continuous. So I know that if this function ...
0
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1answer
34 views

Equivalence of Forms of Baire Category Theorem

I am trying to show the equivalence of two forms of the Baire Category Theorem. These are the two statements: Let $(X,d)$ be a complete metric space. Let $U_n$ be a dense, open set for each $n \in ...
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1answer
24 views

Pick out the correct choices -TIFR 2015

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function and $A \subset \mathbb R$ be defined by $A=\{y \in \mathbb R:y=\lim _{n\rightarrow \infty}f(x_n),$for some sequence $x_n\rightarrow ...