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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Counterexample in metric spaces

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
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Does every metric on a non empty set can be extended on a super set to a metric?

Let $\phi \ne X \subseteq Y$ , let $d$ be a metric on $X$ , then does there exist a metric $d'$ on $Y$ such that $d(x,y)=d'(x,y) , \forall x, y \in X$ ? What if we also assume that the metric $d$ on ...
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Any subset of a metric space is an infinite union of some individual elements of the space?

Let $E$ be a metric space such that the set $\{x\}$ is open $ \forall x \in E$. Does the following proposition make sense? All subsets of $E$ are open. Proof: $\forall S \subset E$, there are ...
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Length structure topology

Let $(X,T)$ be a Hausdorff topological space. Suppose $\emptyset\neq A\subset C(X)$, where $C(X)$ is the set of curves in $X$, that is, the set of continuous maps $\alpha:I\to X$, where ...
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What is the logic underlying this proof?

Proposition: A metric space $X$ is connected if, and only if, every continuous function $f:X\to (\{0,1\},d_D)$ is a constant function, where $d_D$ is the discrete metric on the set $\{0,1\}$. ...
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Prove that two metrics are equivalent

I got stuck on this problem. Hope someone can give some hint to move on. Thanks. Suppose $d_1(x,y) = |x-y|$, $d_2(x,y)=|\phi(x) - \phi(y)|$ where $\phi(x) = {x \over {1 + |x|}}$. Prove that $d_1$ ...
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Does every non-compact bounded metric space support an equivalent metric in which it is unbounded?

Consider $X$ be an infinite set. Let $d$ be a non compact bounded metric on $X$. Can we define an unbounded metric $d'$ on $X$ such that both the metric spaces $(X,d)$ and $(X,d')$ give the same ...
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Let S be a subspace of topological space X. Show that the closure of S, the set of contact points, is indeed closed.

Let $S$ be a subspace of topological space $X$. Show that the closure of $S$, the set of contact points, is indeed closed. I need to prove that the closure is closed but I don't know how to ...
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Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
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Topological spaces without homeomorphisms?

Can we find a topological space which is not homeomorphic to any other? Of course, not considering the space itself neither the empty set. And if's so, is it possible to classify them? Just like the ...
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If an open neighborhood of $x$ has infinite points of $E$, then $x$ is a limit point of $E$

Let $(X, d)$ be a metric space, $E \subseteq X$ and $x \in X \setminus E$. Prove that the following are equivalent: $x \in \overline E$ $x \in \operatorname{Der}(E) = \{x \text{ is an ...
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$\mathbb{R}^2$ to $\mathbb{R}^1$ Injective Mapping While Preserving the Triangle Inequality

Is there a way to map from $\mathbb{R}^2$ to $\mathbb{R}^1$, such that every point in $\mathbb{R}^2$ has a unique point in $\mathbb{R}^1$ and you preserve the distance (isometry) relations of ...
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Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces satisfying the Heine-Borel theorem. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that ...
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31 views

Proving that if $d(x, a) < \varepsilon$ for every $a \in A$, then $d(x, b) \geq \varepsilon$ for every $b \in X \setminus A$

I want to prove the following result: Let $(X, d)$ be a metric space. Then $$\mathring E = \{x \in X \mid d(x, X \setminus E) > 0\}$$ where $d(x, A) = \inf\limits_{y \in A} d(x, y)$. This ...
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Error in proof that the closure of open ball equal the closed ball in all metric spaces

Let $(X, d)$ be a metric space. Denote the open and closed ball as $$B(x_0, r) = \{x \in X \mid d(x, x_0) \lt r\},$$ $$D(x_0, r) = \{x \in X \mid d(x, x_0) \leq r\}.$$ Then $\overline{B(x_0, ...
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Given two metrics $\rho,\sigma$ on $\chi$ show that the following is a metric on $\chi$

The problem asks us to show that $$ \rho_{2}=(\rho^2+\sigma^2)^{1/2} $$ is also a metric on $\chi$. I'm having trouble showing the triangle inequality; I've tried numerous algebraic manipulations ...
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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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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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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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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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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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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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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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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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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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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.