# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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) $d(x,y) = d(y,x)$ ...
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### 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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### 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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### 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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### 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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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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### 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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### 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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Please help me answer the following question: $f: R^n \to R \space is \space continuous \space \iff \space \forall \space \gamma: [a,b] \to R^n \space . \space f \circ \gamma : [a,b] \to R \space is ... 2answers 87 views ### Ultraproduct of a metric space I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin. They start with a complete, intrinsic metric and space$X$and say ... 3answers 97 views ### Do all metric spaces satisfy this property (transitive action by isometries). Do all metric spaces satisfy this property? Suppose$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$. We say a metric ... 1answer 70 views ### Closure of a region and shortest path Let$\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}$exists shortest path (lenth of path is ... 1answer 116 views ### Metric on function space for Pointwise Convergence We say$\{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)$. In this definition we ... 1answer 335 views ### An extension of a continuous function onto the closure Let$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}$. I ... 3answers 33 views ### Area of set-difference Let$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$? 1answer 37 views ### uniform continuity of a function in a metric space Let$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$. 1answer 46 views ### Prove uniform continuity of function I was given$f: <1,+\infty>\times<1,+\infty>\rightarrow <0,+\infty> $defined with$f(x,y)=\ln x+\ln y$and metric on both spaces is induced by taxicab norm. I need to prove this ... 0answers 24 views ### Proving compactness in a geometric scenario Let$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$: $$D = \{(P,Q)|P\in C, Q\in C, ... 1answer 72 views ### distance-measure method to measure the distance between two matrixes(probability distribution) I should find a suitable distance-measure method to measure the distance between two matrixes. The elements of such matrix is 0 to 1, and the sum of the all element is 1, so I think I could treat it ... 0answers 79 views ### Proof of Kuratowski-Wojdyslawski theorem I was reading the Wikipedia page on Kuratowski Embedding, and the following result is stated: The Kuratowski–Wojdysławski theorem states that every bounded metric space X is isometric to a ... 1answer 31 views ### Sequence and Series doubt Suppose x_n \to x in metric space X and y_n \to y in metric space Y. When can we say (x_n,y_n) \to (x,y) ? i.e. what product metric will make it happen ? 1answer 113 views ### For what parameters does a sequence converge in S Let S be space of rapidly decreasing functions f\in C_0^\infty(\mathbb R^n), that for any multi-indices \alpha and \beta there is a constant M_{\alpha,\beta} such that$$|x^\alpha D^\beta f(... 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 ... 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$\...
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 $q \neq p$. ...
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), ...