Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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Complete metric space, with floor function.

I have a problem with this excercise. I need your help. Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ $f(t)=t+[t]$ where $[\cdot]$ is the floor function. Define the metric: $$d(x, ...
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90 views

Consequence of metrizability proof - disregard, the question is an error

In Marian Fabian et al's Functional Analysis and Infinite-Dimensional Geometry, Proposition 3.22 states/proves that if $X$ is a separable Banach space, then the (closed) unit ball, $B_{X^{*}}$ of ...
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127 views

Proving a distance between molecules defines a metric space.

A DNA molecule can be represented as a string of symbols $A$, $C$, $G$ and $T$, such as $$GGATAATTCTAG. . .GACCGTACCC$$ For the purposes of this question, we will assume that all DNA molecules ...
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An inequality for metric spaces: $|d(x, z) − d(y, z)| \le d(x,y)$

Question : Prove $|d(x, z) − d(y, z)|$ is less than or equal to $d(x, y)$. I know I have to use the triangle inequality but I'm just not sure how to apply it with a negative $d(y,x)$.
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75 views

Showing a linear mapping is continuous (or not)

I have three linear mappings: \begin{equation}t_0(f)=f(t_0)\end{equation} \begin{equation}I(f)=\int_{0}^{1}f(t)f_0(t)dt\end{equation} \begin{equation}T(f)=f(t)f_0(t)\end{equation} and I want to ...
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386 views

Is there a non-compact metric space, every open cover of which has a Lebesgue number?

Lebesgue lemma states that for every open cover $\{U_\alpha\}_{\alpha\in A}$ of a compact metric space $(X,\rho)$ there exists a number $d>0$ such that $$ \forall x\in X \quad \exists ...
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130 views

When the set $\{(S_{i_1}\circ\cdots\circ S_{i_n})(x): n\in \mathbb{N},\;\; i_1,\ldots,i_n\in I\}$ is relatively compact?

Let $(X,\rho)$ be a metric space and let $S_1,\ldots,S_N:X\rightarrow X$ be continuous transformations. Denote $I=\{1,\ldots,N\}$. Is it possible to find some minimal assumptions on $S_i$ which would ...
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82 views

Lipschitz continuity for an iterated function system

Let $(M,d_M)$ and $(N,d_N)$ be metric and $$ CB(M)=\{\mbox{all closed bounded subsets of }M\}. $$ Let $f: M\to N$ be a Lipschitz map with Lipschitz constant $L$. Define a map $$ F:(CB(M),\rho)\to ...
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117 views

How to find the Voronoi zone on an infinite plane

Given a plane with an unbounded number of random points, is there an economical algorithm to find the Voronoi zone of any one selected point? I've considered making a "sweeping" circle from that ...
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161 views

Fixed Set Property?

As far as I know, there are fixed-point-like results for continuous functions from a convex compact subset $K$ of an Euclidean space to itself. I have one question in mind: Does there exist a set ...
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162 views

A Closed subset of $M_n(\mathbb{R})$

I can guess that set of Nilpotent Matrices are closed in $M_n(\mathbb{R})$, But I am not able to make it rigorous; I have thought the map $A\mapsto A^k$ is continuous. But then? Please help.
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449 views

Completeness of continuous real valued functions with compact support

How can I show that the space of continuous real valued functions on R with compact support in the usual sup norm metric is not complete ? I know that this result can be proved by using the fact that ...
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188 views

Non-uniform convergence in a compact metric space

$K$ is a compact metric space and we are given a pair of continuous functions f and g: $K \rightarrow \mathbb{R}$ such that f(x) is greater than g(x) for all $x \in X$; Prove that there exists an ...
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Are some discrete (and all finite) metric spaces complete?

For example, it seems to me from the definition of complete that $\mathbb{N}$ with (say) the Euclidean metric would be complete, since any Cauchy sequence on $\mathbb{N}$ must converge to an integer. ...
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231 views

Distance between bounded and compact sets

Let $(X,d)$ be a metric space and define for $B\subset X$ bounded, i.e. $$\operatorname{diam}(B)= \sup \{ d(x,y) \colon x,y\in B \} < \infty,$$ the measure $$\beta(B) = \inf\{r > ...
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274 views

Possible error about properties of boundary points in Simmons's Topology and Modern Analysis

GF Simmons, Introduction to Topology and Modern Analysis Section 11, Pg 68-69 Let $X$ be a metric space and $A$ a subset of $X$. A point in $X$ is called a boundary point of $A$ if each open ...
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Virtually cyclic groups

Let $G$ be a group with finite generating set $A$, define the distance $$d(g,h)=\mathrm{min}\{n:gh^{-1}=a_1^{\varepsilon_1}\dots a_n^{\varepsilon_n}, a_i\in A,\varepsilon_i=\pm1\}.$$ Define the ball ...
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109 views

Prove that $\mathbb{N}$ is nonwhere dense in $\mathbb{R}$

Prove that the set $ \displaystyle{\mathbb{N} =\{1,2,3, \cdots \} }$ is nonwhere dense in metric space $ \displaystyle{ \left( \mathbb{R} ,|\cdot| \right)}$ . I have found a solution in two steps: ...
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96 views

Lipschitz functions and an equality $ f(x) = \inf_y \{ f(y) + kd(x,y) \}$

Let $ f: M \to \mathbf{R}$ a $k$-Lipschitz function, i,e $ |f(x)-f(y)| \le k \cdot d(x,y) $ for every $x,y \in M$. Prove that $ \forall x\in M$ : $$ f\left( x \right) = \inf\limits_{y \in M} ...
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When $(x_n)$ and $(y_m)$ both converge to $x$ then $(y_m)$ is a subsequence of $(x_n)$ if $y_m \in \{x_n: n \in \mathbb{N}\} \cup \{x\}$

I'll state the question first. In any metric space $(X,d)$, assume that $(x_n)$ is a sequence such that $x_n \to x$ for some $x \in X$. If $(y_m)$ is a sequence in $\{x_n: n \in \mathbb{N}\} \cup ...
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119 views

A metric between the closed and bounded sets

Let $M$ be a metric space and consider $Y(M)$ the set of all closed and bounded subsets of $M$. Consider the function $ p:y\left( M \right)^2 \to R $ defined by: $$ p\left( {X,Y} \right) = \max ...
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523 views

Uniformly Continuous Function sending Bounded Set to Unbounded One

Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f: X \to Y$ be a uniformly continuous function. If $A \subset X$ is bounded, must $f(A) \subset Y$ be bounded? It is clear to me that in metric ...
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814 views

diameter on a compact metric space

I have troubles showing the following: Let $(X,\rho)$ be a compact metric space and $F \subset X$ a closed subset. Prove that if diam $F < \infty$, then there exist $x_{0}, y_{0} \in F$ such that ...
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132 views

Best Lipschitz constant

I am trying to find the Lipschitz constant for the following function: $$ f(\pi)=\left|\sum_{i=1}^{m}c_{\pi(i)}-\sum_{i=m+1}^{2m}c_{\pi(i)}\right|, $$ where $c_i \in R$ and $\pi$ is a permutation of ...
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177 views

Cauchy sequences of finite sets

Consider the metric space $\bf R$ with the standard Euclidean metric $d$ and let $F(\bf R)$ denote the collection of all finite subsets of $\bf R$. Endow $F(\bf R)$ with the Hausdorff metric $d_H$. ...
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83 views

A question regarding convergence of distances to closed balls in Banach spaces

Let $X$ be Banach and let $B(x,\varepsilon)$ be the closed ball of radius $\varepsilon>0$ around $x\in X$ and consider the sequence $$f_{n;x}(y)= \begin{cases} 1-n\cdot d(yB(x,\varepsilon)), ...
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221 views

Finite sets are dense with respect to Hausdorff distance

Let $(X,d)$ be a complete metric space and consider \begin{align*} BC(X)&= \lbrace C\subset X\;|\;C\neq\emptyset\text {, closed and bounded} \rbrace\cr \mathrm{Fin}(X)&= \lbrace ...
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105 views

How to preserve completeness between different metrics on the same space?

Let $(M,d)$ be a metric space and $f\colon[0,\infty)\to[0,\infty)$ metric preseving map that is right continuous at $0$, i.e. $f$ has satisfies $$\forall x,y\in [0,\infty)\colon f(x+y)\le ...
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180 views

Measuring closed balls

Let $(X,\parallel \cdot \parallel)$ be Banach and $$\mathcal{BC}(X)=\{A\subset X\colon A \text{ is closed, bounded and non-empty}\}.$$ The natural metric on this space is the Hausdorff distance $d_H$ ...
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Conditions for defining new metrics

Suppose $(M,d)$ is metric. I have proven that if $\psi\colon[0,\infty)\to[0,\infty)$ is non-decreasing, subadditive and satisfies $\psi(x)=0\iff x=0$ for $x\ge0$, then $$\rho(x,y)=\psi(d(x,y))$$ is a ...
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Is the Hausdorff semi-distance Lipschitz?

Let $X$ be Banach (with metric $d$) and let $H(X)$ be the set of closed bounded subsets of $X$. Define for $A,B\in H(X)$ $$\delta(A,B)=\sup_{a\in A}\inf_{b\in B}d(a,b)$$ be the Hausdorff semi-distance ...
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113 views

Convergence of a sequence of functions on closed balls

Let $X$ be a Banach space and $d$ be the induced metric. Let $S(x;r)$ denote the closed ball with radius $r$ at $x\in X$, that is,$$S(x;r)=\lbrace y\in X\colon d(x,y)\le r\rbrace.$$ Let $x,y\in X$ ...
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89 views

Lipschitz mapping on a subset of a metric space

Let $(X,d)$ be a metric space and $Y\subset X$ be a non-empty subset. Is the map given by$$f(x)=\inf\lbrace d(x,y)\colon y\in Y\rbrace$$ a Lipschitz map? And does the equivalence $f(x)=0\iff x\in$ ...
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Why are quotient metric spaces defined this way?

From Wikipedia: If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow the quotient set $M/{\sim}$ with the following (pseudo)metric. Given ...
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358 views

Kuratowski $\limsup$ and $\liminf$ of a sequence of subsets of a topological space?

From Wikipedia: Let $(X, d)$ be a metric space. For any point $x ∈ X$ and any non-empty compact subset $A ⊆ X$, let $$ d(x, A) = \inf \{ d(x, a) \mid a \in A \}. $$ For any sequence of ...
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every isometry is a homeomorphism

We defined an isometry to be a bijection $f:X\rightarrow X'$ such that $d'(f(x_1),f(x_2))=d(x_1,x_2)$ $\forall x_1,x_2\in X$. Show that any isometry is a homeomorphism. So my definition ...
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93 views

Mean value of convergent series

Let us in a normed linear space have a sequence $\{a_i\}_{i=1}^\infty$ which converges to some value $b$, how can I show that $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{a_i}{n}=b$$ My idea is to use ...
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Is a finite union of bounded sets bounded in any metrical space?

In any metrical space $(M,d_M)$, consider $n$ bounded subsets $S_i\subset M$. Then, is $\cup_i^nS_i$ bounded? If so, why?
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Is the complement of a countable set in $\mathbb{R}$ dense? Application to convergence of probability distribution functions.

I am wondering if we have a set $A\in\mathbb{R}$ that is countable, whether $A^{c}$ is dense in $\mathbb{R}$? I thought I saw this quoted somewhere on google but I can not find it again! I am working ...
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Sets where Heine-Borel theorem works [duplicate]

Possible Duplicate: Are there more general spaces than Euclidean spaces to have the Heine–Borel property? By Heine-Borel theorem, a closed and bounded subset of the Euclidean space is ...
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Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
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$X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$

Prove metric space $X$ is connected iff $\forall A\subset X,$ $\partial A\neq\emptyset$. Attempt at a proof: $\rightarrow$ $X$ connected $\implies$ $\forall A\subset X$, $A$ is connected. Then, ...
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309 views

An interval is path connected

$A$ is an interval $\implies$ $A$ is pathwise connected. This kind of goes off one of my previous general questions about path connectedness. I've tried to formalize my attempt at proving this ...
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Tori and metrics

I have been doing some reading on tori. What I can make out of it is that a torus can be equipped with different metrics -- locally Euclidean or as an embedded surface. It is said however that the ...
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490 views

connectedness vs. path connectedness

Is there a general rule of what kind of sets it is easier to prove connectedness using path connectedness or regular connectedness? I understand that path connected $\implies$ connected, but are ...
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How to show $f$ is continuous at $x$ IFF for any sequence ${x_n}$ in $X$ converging to $x$ the sequence $f(x_n)$ converges in $Y$ to $f(x)$

Let $f : (X, d_X) \to (Y, d_Y )$ be a map between two metric spaces. Recall that $f$ is called continuous at $x ∈ X$ if for any open ball $ B_f(x)(ϵ)$ of radius $ϵ$ around $f(x)$ there exists a ball ...
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Show that a subspace $X$ of the Euclidean space $\Bbb{R}^n$ is compact if and only if any sequence of elements of $X$ has a converging subsequence.

Remark: this statement holds in the considerably greater generality of any metric space but the proof of this more general result is quite involved.
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How to show convergence in a metric space?

Suppose that $\{x_n\}→x$ where $\{x_n\}$ is a sequence in a normed space V and $x ∈ V$. Show that $\forall y ∈ V, \{x_n + y\} → x + y$.
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114 views

Disconnectedness on the Real Line

Theorem. If a set $E \subset \mathbb{R}$ is disconnected, then there exist $x, y \in E$, and some $z \in \mathbb{R} \setminus E$ with $x < z < y$. I want to prove this theorem using the ...
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148 views

$A\subset (X,d)$ nowhere dense $\iff$ $(\overline{A})^o=\emptyset$

Show $A\subset (X,d)$ nowhere dense $\iff$ $(\overline{A})^o=\emptyset$. My attempt: $A$ nowhere dense $\implies$ $(\overline{A})^c$ is dense in $X$. Then, $(A^c)^o$ is dense in $X$ (previously ...