Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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Complete space of functions with $f(a)=f(b)$?

Consider M is space of continuous functions (on $[a,b]$) with condition: $f(a)=f(b)$ . Is it complete metric space with $\mu(f,g) =\max\underset{x\in [a,b]}{|(f(x)-g(x))|} $? In my opinion it's true. ...
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1answer
23 views

Homeomorphism between $([0,1]^2, \delta )$ and $[0,1]^2$ with standard euclidean metric

I am stuck as to how I find the homeomorphism described above. $\delta$ is here described to be the metric $$\delta((m_1,n_1), (m_2,n_2))= max\{d_M(m_1,m_2),d_N(n_1,n_2)\}$$ a metric on $M\times N$ ...
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4answers
39 views

How do I visualize a set in metric space?

If I'm given a metric, say the discrete metric $\text d_0(x,y):=\begin{cases} 0, & \text{if }\vec x= \vec y\\ 1, & \text{if } \vec x \neq \vec y\ \end{cases}$ and want to visualize a set ...
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1answer
14 views

Equality of certain distances in a normed vector subspace

Let $E$ be a normed vector space and $F$ a vector subspace of $E$. If $y \in F$, $x \in E$ and $0 < a \in \mathbb R$, prove that $d(y + ax, F) = ad(x,F)$. I've tried to write down the definitions ...
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1answer
24 views

Metric on Lorentzian manifold

Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?
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2answers
34 views

A sequence of continuously differentiable functions with non-differentiable limit

Working in the metric space $C[a,b]$, the set of continuous functions $f:[a,b]\rightarrow[a,b]$, with the supremum metric, I need to demonstrate that $C^1[a,b]$, the subset of continuously ...
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1answer
28 views

A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. The set $\Omega$ is contained in some large disc.

My complex analysis textbook says the following: A set $\Omega$ is bounded if there exists $M > 0$ such that $|z| < M$ whenever $z \in \Omega$. In other words, the set $\Omega$ is contained ...
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1answer
25 views

Obtaining Density From Homeomorphism onto Image

Let $f:X\rightarrow Y$ be a continuous function between metric spaces which is a homeomorphism onto its image and let $K\subseteq X$ be non-empty and $D\subseteq Y$ be dense and satisfy $$ D\cap {f(X)}...
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0answers
66 views

Proof of Schur's Theorem

On Pg. 123 of Schaum's Tensor Calculus: At an isotropic point of $R^n$ the Riemannian curvature is given by $$K=\frac{R_{abcd}}{g_{ac}g_{bd}-g_{ad}g_{bc}}=\frac{R_{abcd}}{G_{abcd}}$$ for any ...
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1answer
76 views

Example of a strong b-metric which is not a metric.

Definition: Let $X$ be an arbitrary set, $d:X\times X\to [0,\infty)$ be a mapping satisfying: (a) $\forall_{x,y\in X}\; d(x,y)=0\iff x=y$; (b) $\forall_{x,y\in X}\; d(x,y)=d(y,x)$; (c) $\exists_{s\...
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1answer
25 views

Metric for the shortest path between two cells of a class I octahedral Goldberg polyhedron.

Consider the following class I octahedral Goldberg polyhedron, $GP_{IV}(9,0)$: Every cell of this polyhedron can be uniquely determined using three coordinates, $x, y, z \in \mathbb{Z}$, such that $\...
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1answer
33 views

Show that there is $x\in[-1;1]$ such as the set of limit point of $u_n(x)$ is $[-1;1]$ where $u_0(x) = x, u_{n+1}(x) = 2u_n(x)^2 - 1$

Let be $f$ defined by $f(x) = 2x^2 -1$. Let be $(u_n(x))$ the sequence defined by: $u_0(x) = x, u_{n+1}(x) = f(u_n(x))$ Show that there is $x\in[-1;1]$ such as the set of limit point of $u_n(x)$ is ...
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0answers
15 views

Relation between total variation and KS distance between measures on $[0,1]^d$

Let $P$ and $Q$ be two probability measures on the space $[0,1]^d$, $d \in \{1, 2, \ldots \}$, endowed with the $L_\infty$ norm and the corresponding Borel $\sigma$-field, $\mathcal{B}$. Let $$F_P(\...
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1answer
26 views

Define a metric on collection of compacts

Let $(\mathbb{R}^2,d)$ be a metric space, where $d$ is the Euclidean metric on $\mathbb{R}^2$. Let $\kappa$ denote the set of compact subsets of $R^2$. Which one of the following expression defines a ...
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0answers
32 views

Prove that $f:X \rightarrow \mathbb{R}$ is constant. [duplicate]

Let $(X,d)$ be a connected metric space and $f:X \rightarrow \mathbb{R}$ be a continuous function. Suppose that for every $x \in X$ there exists an open set $U \subseteq X$ so that $x \in U$ and ${f|}...
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2answers
22 views

Convergence of a sequence in metric space

Can someone please help me with this problem? Thanks! Check if the sequence $x_n= (1+1/n)^n$ is convergent in $ (X,d)$ where $d(x,y)=$ $\frac {2|x-y|}{3+2|x-y|}$, and if it is convergent, then find ...
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1answer
40 views

Metric space which is totally bounded is separable. Baby Rudin Ex 2.24

Let $X$ be a metric space which is totally bounded. Show that $X$ is separable. A metric space is called separable if it contains a countable dense subset. A metric space is called totally bounded ...
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0answers
22 views

How to prove $C_{c}^{\infty}[a,b]$ a complete metric space [duplicate]

How do you prove the space of smooth functions with compact support in an interval $[a,b] \subset \mathbb{R}$, with the metric $\rho(\varphi_{1},\varphi_{2})= \sum_{n=0}^{\infty}2^{-n}\frac{\left \| \...
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1answer
20 views

Is $X$ totally bounded?

I came across the following claim in this post. Let $X$ be a metric space in which every infinite subset has a limit point. Then for every $\delta>0$ there exists $N_{\delta}\in \mathbb{N}$ and $...
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1answer
46 views

Is $C_{c}^{\infty}[a,b]$ a complete metric space?

Is the space of smooth functions with compact support in an interval $[a,b] \subset \mathbb{R}$, with the metric $\rho(\varphi_{1},\varphi_{2})= \sum_{n=0}^{\infty}2^{-n}\frac{\left \| \varphi_{1} - \...
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0answers
16 views

Relationship between Riemann metric and divergence in information geometry

I would like to ask you about formula(6) in this paper. Some people say that the Primary term of the divergence $D$ disappear. But how should I show that?
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1answer
37 views

Uniqueness of completion of a metric space

I've been reading Aliprantis and Burkinshaw. This book gives a nicer construction of completion of metric space(in my opinion) avoiding any equivalence class or pseudometrics. It gives a simple ...
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2answers
43 views

Prove that $x \in \mathbb{R}$ is a limit point of a set $A \subset \mathbb{R}$ if and only if $d(x, A \setminus {x})=0$.

Prove that $x \in \mathbb{R}$ is a limit point of a set $A \subset \mathbb{R}$ if and only if $d(x, A \setminus {x})=0$. I think I have it right but I would like to have it checked. We assume that $...
2
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1answer
28 views

Compact nested subsets [duplicate]

So I am looking over some topology, and am stuck on this problem. I have that $(K_n)_{n\in \Bbb{N}}$ is a nested sequence of non-empty compact sets in a metric sets. I have managed to prove the ...
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0answers
13 views

Constant mapping a contraction?

Is the constant mapping $f:X \rightarrow X$ with $f(x)=K$ on a metric space $(X, d)$ a contraction or a weak contraction ?
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3answers
18 views

Closed subsets of $\Bbb R^n$ that don't verify this property.

I saw this property in an exercise about metric spaces: Let $(E,d)$ be a metric space and $A$, $B$ be two non empty subsets of $E$. If $A$ and $B$ are both compact then there exists $ (a,b) \in A \...
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1answer
25 views

Contraction mapping on metric space

We know the definition of contraction mapping. But it is unkown to me the definition of weak contraction mapping. Help me
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1answer
22 views

Which of the following statements are true?(NBHM-2016,Topology(3.7))

My Try (a)$K\subset M_n(\mathbb R)\sim \mathbb R^{n^2}$ compact iff $K$ is closed and bounded. $A\in K$ has entries from the bounded set. So, eigenvalues are bounded. (b) $tr(A)=1$ is a hyperplane ...
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0answers
48 views

Thin Metric Space

Let $X$ be a complete thin metric space and let $A, B$ be disjoint closed connected subset of $X$ then there is a compact set $K$ such that each neighborhood of $K$ disjoint from $A \cup B$ separates $...
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1answer
33 views

Convergence of measure on clopen basis implies the existence of a limit measure?

Let $X$ be a compact metrizable space. Let $B$ be basis for the topology of $X$. Assume that all sets in $B$ are clopen. Let $(\mu_n)_{n=1}^{\infty}$ be sequence of Borel regular probability measures ...
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0answers
25 views

Two compatible topologically equivalent norms on a module over a non-archimedean field but not metrically equivalent?

The direct question is: Let $F$ be a field, with the trivial norm, that is, $|x|=\begin{cases}1&x\ne0\\0&x=0\end{cases}$. Let $M$ be a free module over $F$ with basis $\left\{x_i\mid i=1,2,...
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2answers
56 views

Boundary Points and Metric space

Definition:The boundary of a subset of a metric space X is defined to be the set $\partial{E}$ $=$ $\bar{E} \cap \overline{X\setminus E}$ Definition: A subset E of X is closed if it is equal to its ...
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1answer
34 views

Open sets in $\mathbb R^2$ and $\mathbb E^2$

Can someone help me with the following problem: Prove that every open set in $\mathbb E^2$ is also open in $(\mathbb R^2, d_1)$, where $d_1((x_1,y_1),(x_2, y_2))=|x_1-x_2|+|y_1-y_2|$ , and vise versa,...
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2answers
32 views

Find the distance between two sets in $\mathbb E^2$

Find the distance between two sets in $\mathbb E^2$: P = {$(x,y): x+2y = 4$} and K = {$(x,y): $$(x+1)^2 + (y+1)^2 = 1$} Need some help with this one
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0answers
98 views

Conclude $\left\|f_n-f\right\|_\infty\to0$ from knowing that $f_n(x_n)\to f(x)$ for all $(x_n)$ and $x$

Let $(X_n,d_n),(Y,d_Y)$ be compact metric spaces for $n\in\mathbb N$, $\pi_n\in C(X_n,Y)$, $$\iota_nf:=f\circ\pi_n\;\;\;\text{for }f\in C(Y)$$ for $n\in\mathbb N$, $f_n\in C(X_n)$ for $n\in\mathbb N$ ...
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2answers
49 views

what is the determinant of [closed]

I'm trying to solve my text book's determinant problem. the determinant is: | 1 x x^3 | | 1 y y^3 | | 1 z z^3 | i have to prove that this determinant ...
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0answers
30 views

Any real valued continuous function on $X$, when restricted on $C$ attains a maximum. If $X$ is compact then prove that $C$ is compact.

Let $C$ be a subset of a compact metric space $(X,d)$. Assume that for every continuous function $h: X\to \mathbb{R}$ , the restriction of $h$ to $C$ attains a maximum on $C$. Prove that $C$ is ...
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2answers
69 views

Complement, open and closed sets

Definition: Let X be a metric space, and E $\subseteq X$, E is closed if it is equal to its closure. Definition: A metric subset U of X is open if for every point in U there exists an open ball ...
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1answer
90 views

If $(E,d)$ is compact and $(f_k)⊆C(E)$ is dense, then $\text E[1∧d(X_n,X)]\to0$ iff $\|f_k(X_n)-f_k(X)\|_{L^1}\to0$ for all $k∈ℕ$

Let $(E,d)$ be a compact metric space and $(f_k)_{k\in\mathbb N}\subseteq C(E)$ be dense. We can show that $$d(x_n,x)\xrightarrow{n\to\infty}0\Leftrightarrow\forall k\in\mathbb N:f_k(x_n)\xrightarrow{...
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1answer
34 views

Interior of a set property

$\DeclareMathOperator{\int}{int}$Let $A,B$ be subspaces of the metric space $X$. If $A \subseteq B$ then $\int A \subseteq \int B$. Proof: Let $x \in\int A$. Since $x$ is interior to the subspace $A$...
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0answers
43 views

If $X$ is a Feller process, then $\sup_{x\in E}\text E\left[d(X_s,X_t)\wedge1\mid X_0=x\right]\xrightarrow{s-t\to0}\to0$

Let $(E,d)$ be a compact metric space, $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on $C(E)$, $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(X_t)_{t\ge0}$ be an $E$...
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0answers
12 views

Using Frenet coordinates as a metric

Having two trajectories, one that is my target and one which is the result of a system trying to follow the target trajectory, I get two slightly different trajectories as a result. The first is my ...
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1answer
63 views

How do p-adic fields degenerate for non-prime $p$?

How do p-adic fields degenerate for non-prime $p$? Let $d(x,y)$ be the inverse of the highest power of $4$ that divides $\lvert x-y\rvert$ Then let $\Bbb Z_4$ be the completion of $\Bbb Z$ under ...
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4answers
72 views

If $K$ is compact and $(f_k)\subseteq C(K)$ is dense, then $x_n\to x$ in $K$ iff $f_k(x_n)\to f_k(x)$ for each $k$

Let $(K,d)$ be a compact metric space and $(f_k)_{k\in\mathbb N}\subseteq C(K)$ be dense (wrt the supremum norm). Let $(x_n)_{n\in\mathbb N}\subseteq E$ and $x\in E$. How can we show that $d(x_n,x)\...
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0answers
117 views

A definition of differentiable functions for arbitrary topological spaces

Background It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I ...
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1answer
37 views

a question about closure of a set in metric space

let $X$ be a metric space and $V \subseteq X$ an open set which seperates two subset $A$ and $B$ of $X$. Then there is an open set $U \subseteq X$ which seperates $A$ and $B$ and such that $\overline{...
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1answer
42 views

How to prove a metric space? [closed]

Suppose $(X, \rho)$ is a metric space. Show that $\forall (x, y) \in X$, $\sigma(x, y) = 2\rho (x, y)$ is also a metric space in X.
3
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2answers
57 views

Compact metric space with “midpoint property” is connected

Let $X$ be a compact metric space. Suppose for every $x$ and $y$ in $X$ there is a point $m$ in $X$ with $d(x, m) = (1/2)d(x, y)$ and $d(y, m) = (1/2)d(x, y)$. Show that $X$ is connected. I'm pretty ...
0
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1answer
16 views

Which properties of $\prod_i E_i$ endowed with $(x,y)↦\sup_id_i(x_i,y_i)$ or $(x,y)↦\sum_i2^{-i}\min(1,d_i(x_i,y_i))$ can we infer?

I know there are similar questions on the board, but they don't answer all of my questions: Let $I$ be a countable set, $(E_i,d_i)$ be a metric space for $i\in I$ and $E:=\prod_{i\in I}E_i$. Can we ...
2
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1answer
60 views

Need reference to continuity of inverse multivalued function.

Let's say $X$ and $Y$ are compact metric spaces. Also, multivalued function $f : X \rightarrow 2^Y$ is continuous in Hausdorff sense. I'm sure it is proven somewhere, that in this case $f^{-1} : Y \...