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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Closure, boundary and interior

Describe the interior, closure and boundary of the following sets in the real line: the set of all integers the set of all rationals the set of all irrationals $(0,1)$ $[0,1]$ ...
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Does such a subset has a nonempty interior?

Let $(a_n)_{n=1}^\infty$ be a sequence such that $0\leq a_n \leq 1$, $\sum_{n=1}^\infty a_n=1$ and let $card \{a_n: n \in \mathbb{N} \}=\infty$. Let's consider the set $$S=\{ \sum_{n\in I} a_n: I ...
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381 views

countable union of proper subspaces

In an interview I was asked to solve a question by using Baire Category Theorem (a complete metric space can not be written as union of nowhere dense subsets), the question was: "Is the vector ...
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118 views

Bounded subspaces and diameters.

Question: Let $X$ be a bounded metric space. Let $Y$ be a subspace of $X$. Prove that $Y$ is bounded and that $\operatorname{diam}(Y) \le \operatorname{diam}(X)$.
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preservation of completeness under homeomorphism

Does homeomorphic metric spaces preserves completeness?I mean two metric space which are homeomorphic and one of them is complete$\Rightarrow$ another one is also complete?
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Union of a connected set and its accumulation point [closed]

Let $A$ be a connected set in the metric space $(X, d)$. If $p$ is an accumulation point of $A$,then prove that $B = A \cup \{p\}$ is connected.
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Metric Space Open Sets.

Let $(X, \rho)$ be a metric. I've shown $\sigma(s,t) = \frac{\rho(s,t)}{1 + \rho(s,t)}$ is also a metric on $X$. I'm having trouble showing that the open sets defined by the metric $\rho$ are the ...
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Integral metric.

In reading I came across the claim that the following is a metric. For the space $X$ of all integrable functions on the interval [$0,1$] , for $f, g \in X$, the following equation defines a metric: ...
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What is the motivation of Levy-Prokhorov metric?

From Wikipedia Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space ...
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Functions in a metric space.

Question Let $(X, d)$ be a metric space. For each $a \in X$, define a function $f_a\colon X \to \mathbb R$ by $f_a(x) = d(x, a), (x ∈ X)$. Prove that for all $a, b \in X$ ...
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197 views

Closed subset of closed subspace is closed in a metric space (X,d)

Is it possible for the following to hold in metric spaces? Let (X,d) be a metric space,if A is closed in Y and Y is closed in X then A is closed in X. If possible someone could assist me for a proof. ...
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Show that for a finite metric space A, every subset is open

Let A be a finite metric space .I want to prove that every subset of A is open. I let the set B, be any subset of A. Since A is finite,then I know that A/B is also finite.I'm stuck here how can this ...
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173 views

Characterizing Open/Closed/compact sets in the metric space $(\mathbb{Z}^n,d)$

What is an open set in the metric space $(\mathbb{Z}^n,d)$, where $d$ is the Euclidean distance in $\mathbb{R}$? As far as I know, in a metric space an open set $O$ is defined as follows: For each ...
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127 views

A new metric involving curves

Let $(X, d)$ be a metric space. The inner metric or length metric associated with $d$ is the function $d_i : X \times X \to [0,\infty]$ defined by $$d_i(x, y) := \inf L(\sigma)$$ where the infimum is ...
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86 views

Show that $(X, d_2)$ is incomplete

I have a set $X = [0, \infty)$ and two metrics: $$ d_1(x, y) = |x-y| $$ $$ d_2(x, y) = \left| \frac{x}{1+x} - \frac{y}{1+y} \right| $$ I already showed that $d_1$ is equivalent to $d_2$. Now I have ...
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105 views

Curves and geodesics

This is a very long problem of homework. Definitions: We start by defining a curve as a continuous function $ \phi :\left[ {a,b} \right] \to \left( {M,d} \right) $ where M is a metric space with ...
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Describe and illustrate the ball $B_1(0,0) $.

On $\mathbb{R}^2$ we have a metric defined by $d(x,y)=|x_1- y_1|+ |x_2- y_2|$. Describe and illustrate $B_1(0,0)$, the ball of radius $1$ centered at the origin $(0,0)$. SOLUTION By definition ...
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Example of a homeomorphic map $T:X→Y$

Definition. Let $X$,$Y$ be metric spaces.Then a map $T:X\to Y$ is an homeomorphism if $T$ is continuous, open and bijective. I don't find a counterexample of such maps, may someone give me at least ...
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Showing that $d(m,n)=|m^{-1}-n^{-1}|$ is not a complete metric on $\mathbb Z^+$

Let $X$ be a set of all positive integers and define metric $d$ on $X$ by $d(m,n)=|m^{-1} - n^{-1}|$. I'm required to show $(X,d)$ is not a complete space. SOLUTION: Let $\{x_n\}$ be any Cauchy ...
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To show $X_2$ is complete space

Suppose $X_1$ and $X_2$ are isometric and $X_1$ is a complete space; show that $X_2$ is a complete space. Here I need somebody to help me or to give me ideas.
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139 views

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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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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132 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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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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439 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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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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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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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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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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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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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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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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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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290 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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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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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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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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552 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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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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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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84 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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239 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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106 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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182 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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179 views

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 ...