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 dense set in plane

Is there a dense set in $\Bbb{R^2}$ that every vertical line or horizontal line intersect in finite points. I think that we can consider $\Bbb{Q} ×\Bbb{Q}$ but every vertical line or horizontal line ...
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How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
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Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?

I am curious about whether $|x|^p$ with $0<p<1$ satisfy $|x+y|^p\leq|x|^p+|y|^p$ for $x,y\in\mathbb{R}$. So far my trials show that this seems to be right... So can anybody confirm whether ...
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Constructing a metric $\rho$ such that $(\mathbb{R}\setminus \{0\},\rho)$ is a complete metric space

Let $S = \mathbb{R}\setminus \{0\}.$ Construct a metric $\rho$ on $S$ such that (1) $(S,\rho)$ is a complete metric space and (2) for any sequence $\{s_n\}$ in $S$ and $s \in S,$ the ...
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Is the space of subsets of $\mathbb R^n$ with the Hausdorff metric separable?

Let M be the Metric Space whose "points" are the Closed and Bounded subsets of a finite dimensional Euclidean Space and whose "distance function" is the Metric defined by Hausdorff for such point ...
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Root distance function in Metric space [duplicate]

Let $\mathbf X = \Bbb R$ with distance function defined by $d(x,y) = {|x-y|}^\alpha$ , where $\alpha \in \Bbb R$ $(0<\alpha\le1)$. Prove that $(\Bbb R , d)$ is a metric space. The first three ...
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distribute K points in N dimensional space

I'll try to do my best to simplify the problem, I'm not a Mathematician, I'm a Computer Engineering Student. I'm doing the K-means algorythm, for those who doesn't know what is, is an algorythm to ...
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If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
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61 views

The measures used to define Hausdorf dimension versus Haar measure

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$ $$H_\delta^\alpha ...
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1answer
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Continuous functions on a closed subset of a topological space

Let $X$ be a topological space with $Y$ a closed subspace with relative topology. If $f:Y \rightarrow Z$ is a continuous map of topological spaces, then can $f$ always be extended to be from $X$ to ...
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21 views

Show that $d_2$ is not a metric.

Show that the function $d_2$ given by $d_2(f_1, f_2)^2 = \int_a^b{(f_1 - f_2)^2}$ is not a metric space on the space of Riemann integrable functions on $[a,b]$. $d_2(f_1, f_2) = 0$ iff $f_1 = ...
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simple proof for principle of pigeons

I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use: Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$ The principle of the pigeons ...
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1answer
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Should a metric always map into $\mathbf{R}$?

Typically you see the definition of a metric as a function which maps $X\times X\to\mathbf{R},$ but does this always have to be the case? Motivating example: When you complete $\mathbf{Q}$ with the ...
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Example of a complete metric space which is not compact

Is there any example of a complete metric space which is not compact? Why?
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Example of a bounded space which is not totally bounded

I was trying to find an example of a bounded metric space which is not totally bounded. The only example I could come up whith was the natural numbers with the discrete metric. However, like any ...
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1answer
12 views

Dense subset in which Cauchy sequences are convergent

Let $S$ be a dense set of a metric space $X$, such that all Cauchy sequences in $S$ are convergent (not necessarily in $S$). Then $X$ is complete space. How can I show that $X$ is complete space ...
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Computationally efficient means of determining distance in the Skorohod Topology?

I have two functions f and g in a computer. Domain 1...N. I'd like to compute their distance using the Skorohod Topology in an efficient manner. (I first ran across this metric many years ago in ...
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18 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
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1answer
16 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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5answers
956 views

Is a ball always connected in a connected metric space?

If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
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If $F$ is closed subset of $R^n$ and $x \in R^n, $ is $x+F$ still closed? [on hold]

If $F$ is closed subset of $R^n$ and $x \in R^n, $ is $x+F$ still closed ?
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Proving that $B:=\{f(x)\in C[a,b]:f(a)=0\}$ is close set

Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$ My attempt: Metric space $C[a,b]$ ...
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Puzzled with this number theory/analysis problem

So, I am having this problem, let $N(x,y)$ be the greatest integer which $b^{N(x,y)}|x-y$ where $x,y$ are integers in $\mathbb{Z}$. Assume that $b \geq 2$. Show $d(x,y)=b^{-N(x,y)}$ is a metric. ...
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Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
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Cauchy sequence of natural numbers

Consider the set consisting of all cauchy sequences $a_n$ with $a_n \epsilon \mathbb{N}$ for all $n$. Is the set countable? My idea: It is straight forward to prove that any such cauchy sequence ...
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If X is compact and $C(X)$ is the space of all continuous real valued functions. Prove $C(X)$ is a complete metric space.

Let $X$ be a compact metric space and define $C(X)$ to be the space of all continuous real valued functions on $X$ with a metric defined by $$d(f,g)=\sup_{x \in X} |f(x) -g(x)|.$$ Show that $C(X)$ is ...
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Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...
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25 views

Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
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Metric space of non empty closed bounded parts of $R$ with the Hausdorff metric

Consider the metric space of non empty closed bounded parts of $R$ with the Hausdorff metric. For n $\in N_{0}$ and $F_{n} = \{0,1/n,2/n,3/n, ..., 1\}$ i am wondering if $(F_{n})_{n}$ is convergent? ...
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subset of C([0,1]) limited in d1 metric but not in d_inf metric

I am wondering if it is possible to find a subset $C(X) = C([a,b])$ of $C([0,1])$ which is limited for the d1 metric $(d_1(f,g) = \int_{a}^{b} |f(x)-g(x)|dx)$ but not for the $d_{inf}$ metric ...
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Complete metric space

Let $C^0([a,b])$ denote the space of continuous function $f:[a,b]→\Bbb R$. Define $d(f,g)=sup_{[a,b]}|f-g|$. I've proved that d is metric in $C^0([a,b])$. How to prove that this metric space is ...
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Cover $(0, +\infty )$ by open sets

Cover $(0, +\infty)$ by open sets $U_\alpha$ such that for any $\epsilon > 0$ there are points $x, y \in (0, +\infty)$ with $|x-y|<\epsilon$, not both belonging to the same $U_\alpha$ The ...
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A point in a closed set in Euclidean Space [duplicate]

''There exists a point in a closed set which is at minimum distance from a point not in the set.'' I have no idea why this is true. Any help will be appreciated.
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Distances between closed sets on metric spaces

Which says that $\mathbb R^n$ the distance between a point $b$ and a set $X$ defined by$$ \inf \left\{ d(b,x) \mid x \in X \right\} $$ The proposition: If $X$ is closed, this distance is reached ...
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If $d_1$, $d_2$ are metrics on $X$ find a relationship between $\tau_1$ and $\tau_2$.

Suppose $d_1$, $d_2$ are metrics on $X$ and whenever $x_n \rightarrow x$ using $d_1$ we have that $x_n \rightarrow x$ using $d_2$. Let $\tau_1$ be the collection of open sets of $(X,d_1)$ and ...
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Suppose $d_1$ and $d_2$ are equivalent metrics and $d_1$ is bounded, is $d_2$ bounded?

Suppose $d_1,d_2$ are topologically equivalent metrics on a set $X$. Suppose also that $d_1$ is bounded, that is there exists $K>0$ such that $d_1(x,y) \leq K$ for all $x,y\in X$. Does this mean ...
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Prove this is a metric, what else should I consider?

Let $C_b(\mathbb{R})$ be the space of the bounded continuous functions with values in $\mathbb{C}$ defined in $\mathbb{R}$ ($f:\mathbb{R}\rightarrow\mathbb{C}$) prove that: with $x\in \mathbb{R}, ...
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Metric space and continuity

We define a map $f:(S,d)→(S',d')$ between 2 metric spaces to be continuous at x belongs to S if for every sequence ${x_n}$ in $S$ that converges to x, the sequence {f(x_n)} in $S'$ is convergent to ...
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Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in ...
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Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C_b[0,1]$

Following Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ I would like to prove that the same is true for bounded functions on $[0,1]$ ...
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Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.

I have some questions about this proof that "Every open ball of a normed vector space $E$ its homeomorph to the entire space $E$.": By the example (12), we just have to consider the ball $B(0,1)$, we ...
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How is $d(af(x), af(x_o))$ and $d(f(x), f(x_o))$ related?

I wish to prove that given $f \in C_0([0,1])$ of continuous function, then $af \in C_0([0,1])$ where $a \in \mathbb{R}$ I am having trouble relating $d(af(x), af(x_o))$ with $d(f(x), f(x_o))$ So to ...
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There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
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Is it possible that two metric spaces are metrically isomorphic but not homeomorphic.

I am trying to find an example of metric spaces $(X,d_x)$ and $(Y,d_y)$ such that they are metrically isomorphic, but not homeomorphic. I have been attempting to find one, however I have not been ...
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If $d_1,d_2$ are not equivalent metrics, is it true $(X,d_1)$ is not homeomorphic to $(X,d_2)$?

Consider the statement: If $(X,d_1)$ and $(X,d_2)$ are metric spaces and $d_1,d_2$ are not equivalent metrics, then $(X,d_1)$ is not homeomorphic to $(X,d_2)$. I think this is true, however I can't ...
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Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
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A map between metric spaces preserving convergent sequences is continuous

Pugh, "Mathematical Analysis", exercise 2.17: Assume $f : M \to N$ is a map from one metric space to another which satisfies the following condition: for every convergent sequence $(a_n) \subset ...
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How to prove that the uniform topology is different from both the product and the box topology?

Let $J$ be an arbitrary index set. Then how to prove that the uniform topology on the Cartesian product $\mathbf{R}^J$ of the set $\mathbf{R}$ of real numbers with itself is different from both the ...
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Normalized measure over compact metric spaces

Consider the following definitions. Let $M = (V,T,d)$ be a compact metric space with finite diameter $$D = D(M) = \max d(x,y), ( x, y \in M)$$ and a finite normalized measure $\mu$$M$(.), ...
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Distance from a point to empty set.

Let $(X,d)$ be a metric space and let $A \subseteq X$. We define the distance from a point $x \in X$ to $A$ by $d(x,A)= \inf \{ d(x,a) : a \in A \} $. What will be the value of $d(x, \emptyset )$? I ...