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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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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18 views

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
95 views

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 of ...
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2answers
94 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 $X+Y$ ...
3
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2answers
106 views

Proof that the function $f(x)=d(x,y)$ 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
67 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
75 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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2answers
112 views

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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1answer
109 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) $d(x,y) = d(y,x)$ ...
4
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3answers
209 views

condition for equivalence of norms on vector spaces

Let us call two norms $|x|_1$ and $|x|_2$on a finite-dimensional vector space equivalent if they set the same topology on that space. I need to show that this definition is equivalent to the existence ...
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1answer
46 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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0answers
27 views

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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3answers
934 views

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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1answer
238 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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2answers
44 views

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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1answer
95 views

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
70 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 $(x_k)\...
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2answers
75 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 d'...
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1answer
21 views

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 $d(x,y)=\sup\{|x(t)-y(t)|:t\in(-\infty,\infty)...
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780 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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180 views

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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1answer
237 views

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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143 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 $d(A,B):=$max$\{$min$...
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1answer
509 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 distance ...
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71 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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1answer
119 views

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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38 views

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
63 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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67 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 not....
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1answer
35 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 D(v,3)...
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48 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
58 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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87 views

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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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 ...
2
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1answer
70 views

Closure of a region and shortest path

Let $\Omega$ be a region in $\mathbb{R}^2$ and $\overline{\Omega}$ be closure of $\Omega$. Is it true that between every two points $x,y \in \overline{\Omega}$ exists shortest path (lenth of path is ...
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1answer
116 views

Metric on function space for Pointwise Convergence

We say $\{f_n:[a,b]\to \mathbb{R}\}$ is a sequence of functions converging pointwise to $f:[a,b]\to \mathbb{R}$ if for any $x$ in $[a,b]$ we have $f_n(x)\rightarrow f(x)$. In this definition we ...
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1answer
335 views

An extension of a continuous function onto the closure

Let $S$ be a proper subset (not closed) of a metric space $X$. Suppose that $f:S \to \Bbb R$ be continuous. I want to know the condition under which $f$ can be continuously extended to $\bar{S}$. I ...
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Area of set-difference

Let $X$ and $Y$ be two open sets in $\mathbb{R}^2$, with $X\subsetneq Y$. Is it possible that $\text{Area}(Y\setminus X)=0$? Is it possible that $\text{Area}(Y\setminus Closure[X])=0$?
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1answer
37 views

uniform continuity of a function in a metric space

Let $A\neq\emptyset$ be a given subset of a metric space $(X,d).$ If $f(x)=d(x,A)$ show that $f$ is uniformly continuous on $X$.
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1answer
46 views

Prove uniform continuity of function

I was given $f: <1,+\infty>\times<1,+\infty>\rightarrow <0,+\infty> $ defined with $f(x,y)=\ln x+\ln y$ and metric on both spaces is induced by taxicab norm. I need to prove this ...
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Proving compactness in a geometric scenario

Let $C$ be a compact subset of $R^2$. Let $D$ be the set of all pairs of points $(P,Q)$ from $C$, such that the open segment between $P$ and $Q$ is contained in $C$: $$D = \{(P,Q)|P\in C, Q\in C, ...
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1answer
72 views

distance-measure method to measure the distance between two matrixes(probability distribution)

I should find a suitable distance-measure method to measure the distance between two matrixes. The elements of such matrix is 0 to 1, and the sum of the all element is 1, so I think I could treat it ...
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79 views

Proof of Kuratowski-Wojdyslawski theorem

I was reading the Wikipedia page on Kuratowski Embedding, and the following result is stated: The Kuratowski–Wojdysławski theorem states that every bounded metric space $X$ is isometric to a ...
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1answer
31 views

Sequence and Series doubt

Suppose $x_n \to x$ in metric space $X$ and $y_n \to y$ in metric space $Y$. When can we say $(x_n,y_n) \to (x,y)$ ? i.e. what product metric will make it happen ?
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1answer
113 views

For what parameters does a sequence converge in $S$

Let $S$ be space of rapidly decreasing functions $f\in C_0^\infty(\mathbb R^n)$, that for any multi-indices $\alpha$ and $\beta$ there is a constant $M_{\alpha,\beta}$ such that $$|x^\alpha D^\beta f(...
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1answer
84 views

Adherent values for a sequence in a metric space

I have these two definitions for an adherent value of a sequence the first is : $a$ is a an adherent value for $(x_n)$ iff $$\displaystyle \forall \varepsilon>0,\forall n\in \mathbb{N},\exists ...
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1answer
152 views

A problem in A Course in Point Set Topology by Conway, union of totally bounded sets

This is stated as a problem in A Course in Point Set Topology book by J. Conway: Let $\{E_n\}$ be a sequence of totally bounded sets. If $\operatorname{diam}E_n\to 0$ as $n\to\infty$, show that $\...
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2answers
157 views

Negating the definition of a limit point

Below is a definition of a limit point: $E$ is a subset of a metric space $X$. $p \in X$ is a limit point of $E$ exactly when every ball around $p$ has an element $q \in E$ such that $q \neq p$. ...
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1answer
166 views

Jingle River (Berbed wire) Metric Problem

I want to prove that Jungle River metric is indeed a metric space, and determine it is open and closed balls. Firstly, i know that the metric is given by $x,y\in \mathbb{R}^2$, such that $x=(x_1,x_2), ...