Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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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
66 views

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

$(X,d)$ 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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1answer
35 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
59 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 ...
3
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1answer
157 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
344 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. ...
2
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1answer
171 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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82 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
58 views

$X$ 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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0answers
46 views

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 $R_{00}$, ...
4
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1answer
55 views

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
119 views

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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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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2answers
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If $f$ is continuous & $\lim_{|x|\to {\infty}}f(x)=0$ then $f$ is uniformly continuous or NOT?

Let, $f:\mathbb R\to \mathbb R$ be a continuous function such that $\lim_{|x|\to {\infty}}f(x)=0.$ Then prove or disprove that $f$ is uniformly continuous. I tried through the formal definition ...
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91 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 $X+Y$ ...
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2answers
80 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 ...
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1answer
63 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
70 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 ...
2
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1answer
49 views

To find a counterexample in metric space.

Suppose $X$ is a metric space, $z \in X$ and $(x_n)$ is a sequence in $X$. Show that if $X$ has a subsequence that converges to $z$, then dist$(z ,$ {$x_n :n ∈ N$}) $= 0$, and show also that the ...
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confusion over Finite intersection property

It is stated that $A_n={(\frac{-1}{n},\frac{1}{n})}$, then arbitrary intersection of open sets need not be open is true as in this case $\bigcap_{i=1}^{\infty}=\left \{0 \right \}$ is not open. Now ...
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82 views

Showing $d(x,y) = \frac{|x-y|}{1+|x-y|}$ is a distance.

Show that $(\mathbb{N}, d)$ is a metric space with $$d(x,y) = \frac{|x-y|}{1+|x-y|}$$ My attempt: let $x,y \in \mathbb{N}$, 1) $d(x,y) = 0 \implies |x-y| = 0 \iff x = y$ 2) $d(x,y) = ...
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1answer
41 views

Distance to a closed ball in a normed space.

Let $(E, \|\cdot\|)$ be a normed vector space, and consider $B = B[{\bf a},r]$ the closed ball. Let ${\bf b}\in E$. Then $\newcommand{\d}{{\rm d}} \d({\bf b},B) = 0$ if and only if ${\bf b} \in B$. ...
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What condition is needed on $S$ and $T$ such that $C(S,T)$ be compact.

If $C(T,S)$ is the set of all continuos function between $T$ and $S$ metric spaces and $S$ compact with the uniform metric. What conditions are needed on $T$ and $S$ such that $C(T,S)$ be compact? ...
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Proving that the triangle inequality holds for a metric on $\mathbb{C}$

Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as: $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} + \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$ I ...
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150 views

Check if the given set is Connected and Compact.

$S=\left\{\dfrac{x^{2}}{1+x^{2}}:x \in \mathbb R\right\}$ Since $S$ is not closed (the limit point $1$ does not belong to the set), so I concluded that $S$ is not compact. I am confused about ...
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Equivalent matrics

Let $ (X,d) $ be a metric space and let $f:[0,\infty)\to [0,\infty)$ be a continuous function with the following properties: (i) $ f(x)=0 $ iff $x=0$. (ii) $ f(x)\leq f(y) $ if $ 0\leq x\leq y $. ...
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Show that the function $d(x, y)$ is a metric on the set $\mathbb R^2$ . [closed]

Show that the function $d(x, y) = |x_1 − y_1| + |x_2 − y_2|$, where $x = (x_1, x_2), y = (y_1, y_2)$, is a metric on the set $\mathbb R^2$ . I have question about metric spaces from topology. Can ...
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1answer
69 views

Is it a closed set?

I'm in the metric space $(\mathbb{R},d)$ where $d(x,y)=\frac{|x-y|}{1+|x-y|}$ and i have to prove that $I_n=[n,+\infty[, n\in \mathbb{N}^*$ is closed, is it right to take a convergent sequence ...
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69 views

Metric equivalence

I have that $E=[0,1]$ and $d'(x,y)=|\sqrt{x}-\sqrt{y}|$ and i want to prove that $d'$ and the usual metric $d(x,y)=|x-y|$ are not equivalent in the metric sense. I proved easely that $d(x,y)\leq 2 ...
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1answer
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Separable spaces, need help

Please if someone could tell me how to proof this below. Problem: Is the space of continuous and bounded functions on $(-\infty,\infty)$, with the metric ...
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537 views

Proof that any finite subset of a metric space is closed

I have a metric space $(X, d)$ and I am trying to prove that any finite subset $F = \{x_1,\ldots,x_n\} $ of $X$ is closed. What I have by now is a proof that a subset $F$ of a metric space $X$ is ...
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Given a finite metric space, are the matrices of triangle inequality errors invertible?

I have been working on some problems regarding finite metric spaces and have already proven/positively answered the following statement/question if the underlying metric has additional properties. Now ...
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non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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123 views

Connectedness of a metric space implies connectedness of the corresponding Hausdorff hyperspace

If $(X,d)$ is a connected metric space, show that $(\mathscr H(X),h)$ is also connected. where h is the Hausdorff distance, define by max$\{d(A,B),d(B,A)\}$ and ...
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1answer
347 views

Prove there is a minimum distance between a closed and compact sets.

Let $A$ be a compact set and $B$ a closed set ($\varnothing\ne A,B\subseteq \mathbb{R}^n$). Prove there's a minimum distance between $A$ and $B$. In class we've seen that there's a minimum ...
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66 views

Question about closed set

Please if someone could give me an answer to this problem: Show that $A=\{x \in l_2:|x_i| \le \frac 1 i, i=1,2,3,\ldots\}$ is a closed set in $l_2$. Where $l_2$ is the set of sequences in $\Bbb R$ or ...
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Completion of a Banach space with respect to a different norm

Let $(X,|\cdot|_X)$ be a Banach space. Define a space $Y$ as the completion of $X$ under a norm $$|u|_Y = |u|_X + |Tu|_Z$$ where $T:X \to Z$ is a linear continuous map where $X \subset Z$ is a ...
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Which of the following spaces are complete

Is the following space complete? $X_1=\left(0,\dfrac{\pi}{2}\right)$ defined by $d (x,y)=|\tan x-\tan y \ |$ Let $x_n$ be a Cauchy sequence in $X$ then, we will have $n,m\in \mathbb N$ such that ...
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1answer
58 views

How to understand point functions

I am having trouble understanding the meaning of point functions. I know the mathematical definition but i don't think that i truly understand there true meaning. Point functions: Suppose $(X,d)$ is ...
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64 views

Determine completeness of a metric space

Let $X := (0,\infty)$ and $\left(X, \rho: X\times X\rightarrow (0,\infty), (x,y)\mapsto\rho(x, y):=\left|\frac{1}{x}-\frac{1}{y}\right|\right)$ a metric space. Determine whether it is complete or ...
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1answer
24 views

problem on union of connected sets

for $v \in \mathbb R^2$ and $r>0$ let $D(v,r)$ denote the closed disc with center at $v$ and radius $r$ let $v=(5,0)$ for $\alpha >0$ let $X_\alpha $ be the subset $X_\alpha =D(-v,3)\cup ...
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46 views

Need help in metric spaces proving this statement!

Please if someone could help me prove this rather annoying statement. Let $C(0,1)$ be the set of continuous functions on the open interval $(0,1) \subset \mathbb R$. Fro any two functions $x(t), y(t) ...
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1answer
54 views

function is continuous iff its composition with a curve is continuous

Please help me answer the following question: $ f: R^n \to R \space is \space continuous \space \iff \space \forall \space \gamma: [a,b] \to R^n \space . \space f \circ \gamma : [a,b] \to R \space is ...
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Ultraproduct of a metric space

I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin. They start with a complete, intrinsic metric and space $X$ and say ...
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86 views

Do all metric spaces satisfy this property (transitive action by isometries).

Do all metric spaces satisfy this property? Suppose $A$ and $B$ are finite sequences $a_1,a_2\dots a_n$ and $b_1,b_2\dots b_n$ such that $d(a_i,a_k)=d(b_i,b_k)$ for all valid $i,k$. We say a metric ...