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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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Compute $\|T\|$ when $D (T)$ is endowed with $\|\cdot \|_{\infty}$ norm

Let $X=\{f\in C[0,1]:f (0)=0\}$. Define $$T:X \to \mathbb R$$ $$T_f=\int_0^1 f (t)dt$$ Compute $\|T\|$ when $X$ is endowed with $\|\cdot\|_{\infty}$. By approximating, I have $$|T_f| \leq\int_0^1 |f (...
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Proof that the metric space $M$ is complete if every closed ball of $M$ is complete.

Let $M$ be a metric space I'm asked to prove the statement "Every closed ball of $M$ is complete $\implies$ $M$ is complete". My attempt at this is as follows: Let $\{y_i\}$ be a cauchy ...
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Intrinsic vs. extrinsic surface curvature

This may be just a terminology question (or not). A 2-cylinder is intrinsically flat. Its curvature cannot be detected from inside (although its topology can be studied by making various round trips)....
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Metrics on functions that use both the domain and range's metric.

If $(X, d_x)$ and $(Y, d_y)$ are metric spaces, then the metric on functions from $X$ to $Y$ is often defined as their supermum difference in $Y$. However, this only uses the metric on $Y$. Can you ...
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Proving that $Y$ is complete

Let $X$ be a dense subset of $Y$. Let every cauchy sequence in $X$ converge to a point in $Y$ $(1)$. By the definition of dense I know that every point in $Y$ is either in $X$ or is a limit point ...
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Approximate monotonicity of $\epsilon$-covering number

This is from Exercise 4.2.10 in Roman Vershynin's book, High-Dimensional Probability: An Introduction with Applications in Data Science. Let $(T;d)$ be a metric space and $N(A,d,\epsilon)$ be the $\...
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Graph theory and compact metric spaces: is it possible?

Is there a theory to study mathematical objects that are graphs but being its nodes higher dimensional compact metric spaces? Thanks!
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Then $X$ is compact , path connected and connected set ?? [on hold]

$X$ be a subset of $\mathbb{R}^{2}$ $X=\{(x,y)|x=0,|y|\leq 1\} \cap \{(x,y)|0<x\leq 1 ,y=\sin(\frac{1}{x})\} $ Then $X$ is compact and path connected?? How to prove its compact I think x=0, y co-...
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Is there a distance preserving map from the metric $d(a,b) = \log(|ab| + 1)$ to a Euclidean metric?

Let $M$ be a metric space, where the point are points in the euclidean plane and $$d_M(a, b) = \log(|ab|+1)$$ where $|ab|$ is the euclidean distance from $a$ to $b$. This is a metric since $x \mapsto \...
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A compact subset of a metric space is always closed

I am reading Principles of Mathematical Analysis by Walter Rudin. In chapter two, it has the following definitions: A neighborhood of $p$ is a set $N_r(p)$ consisting of all $q$ such that $d(p,q) \...
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some confusion about singleton set?

Is the set $\{0\}$ is closed in $(\mathbb{R} , |.|)$ ? where $|.|$ denotes the usual metric on $\mathbb{R}$ My attempt : yes , because i think $\{0\}$ is not open for all $x> 0, (x- \...
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Function spaces on equivalent metric spaces

Suppose $X$ and $Y$ are two complete metric spaces, and let $\Phi:X\to Y$ be a bi-lipschitz map. For $\alpha>0$, denote $\mathcal{C}^\alpha(X)$ (resp. $Y$) the set of functions $f:X\to\mathbb{R}$ ...
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Closed sets with filters

I am studying topological vector spaces, and, being at the begin of the textbook (Sevres) I frequently encounter rephrasing and generalisations of known results about metric spaces in terms of filters....
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Which surfaces can be isometrically embedded in a flat 3-space?

In 1971 Jacobowitz proved that it is always possible to locally isometrically embed a surface in $\mathbb R^4$ (thanks to @doetoe for the reference). Not all surfaces can be locally isometrically ...
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Embedding for homogeneous Sobolev spaces

Let $B_R$ be a ball of radius $R$ in $\mathbb{R}^d$. Under what conditions does the Sobolev embedding $\dot{W}^{1,p}_0(B_R) \hookrightarrow L^{p^*}(B_R)$ holds, where $p^* = \frac{np}{n-p}$? Here $\...
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Supremum metric on closed subset of continuous functions

I have a question that goes as follows: Consider the $\sup$ metric $$d(f,g) = \sup_{x\in[a,b]} |f(x)-g(x)|$$ Let $\mathcal{C}[0,1]$ be the space of continuous real-valued functions on $[0,1]$ ...
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Metric topological manifold and the choice of neighbourhood

Let $(M,d)$ be a metric topological manifold (without boundary). We know that for any $m\in M$ there is an open neighbourhood $U\subseteq M$ of $m$ such that $U$ is homeomorphic to $\mathbb{R}^n$. Can ...
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57 views

Is it true that every closed set is a countable intersection of open intervals?

We know that every open set is a countable union of open intervals with rational endpoints and that every open interval is a countable union of closed intervals. Hence every open set is a countable ...
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Uncountable set with a non-discrete metric [on hold]

Let $X$ be an uncountable set. Can we always define a metric on this set such that $X$ has a metric topology which is not discrete?
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If $T^m$ is a contraction, has $T$ got a fixed point? [duplicate]

Let $(X,d)$ be a metric space and complete and $T: X\to X$ a mapping such that $T^m$ is a contraction. Show that $T$ has a unique fixed point. It is clear that $T^m$ has a unique fixed point (Banach ...
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Let (X,d) a metric space. Show that a function is continuous [duplicate]

Let (X,d) a metric space. Show that the function y↦d(x,y) is continuous. Please help me, I don't know where to start.
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exact definition of open sets [on hold]

What is the exact definition of open sets? I know both definitions of open sets with respect to metric space and topological space. but what is most general definition that covers the both.
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Theorem 2.34 Baby Rudin (Compact subsets of metric spaces are closed): How is his proof general enough?

2.34 Theorem Compact subsets of metric spaces are closed. Proof. Let $K$ be a compact subset of metric space $X$. Let $p\in K^c$, $q\in K$. Let $V_q, W_q$ be neighborhoods of $p$ and $q$ ...
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$M$ is metric space. $A\subset M, F$ is closed in $M$, prove $A\cap F$ is closed in $A$ [duplicate]

$M$ is metric space, $A\subset M, F$ is closed in $M$, I'm asked to prove $A\cap F$ is closed in $A$. First of all, what does "closed in $A$ mean"? Does it mean that $A$ is now the metric space ...
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Quotient Space Metric with Nice Equivalence Classes

Quotient Space Metric: The quotient metric for arbitrary quotient spaces is defined as If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow ...
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Is this proof that $\text{diam}(A) = \text{diam}(\bar{A})$ correct?

Let $M$ be a metric space. I'm asked to prove that the diameter of a set $A\subset M$ is the same diameter as its closure $\bar{A}\subset M,$ $$\text{diam}(A) = \text{diam}(\bar{A}).$$ My attempt: ...
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1answer
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Prove that this group action is continuous from $S\times\Bbb{R}^2\to\Bbb{R}^2$

Let me give some description about the notations used- S denotes the collection of all transformations on $\Bbb{R}^2$ by integer coordinates. i.e. $S=\{t_v|t_v(x)=x+v,\forall x\in\Bbb{R}^2; \forall v\...
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Prove the equivalent conditions for nowhere dense subset.

Let $(X,d)$ be a metric space and $A$ be a subset of $X$. Then the following statements are equivalent. $A$ is nowhere dense. $\overline{A}$ doesn't contain any non-empty open set. Each ...
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Preimage of continuous one-to-one function on connected domain is not continuous.

I know that given $B$, a compact subset of $\mathbb{R}^n$, and $f : B \to \mathbb{R}^m$, a continuous injective (one-to-one) function, $f^{-1}$ is continuous on $f(B)$. (This true). I also know that ...
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Show that $\operatorname{Cl}(A)\setminus A$ consists of entirely of accumulation points of $A$

Let $(M,d)$ be a metric space and $A \subset M$. Show that $\operatorname{Cl}(A)\setminus A$ consists of entirely of accumulation points of $A$, where $\operatorname{Cl}(A)$ is the closure of $A$. My ...
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$x \neq y\in$ metric space $M$, prove $\exists$ open sets $U,V$ s.t. $x\in U,\ y\in V$ and $\bar{U} \cap \bar{V} = \emptyset$

Let $x,y\in M, \ x\neq y,\ M \ \text{being a metric space}$. I'm asked to prove that there exists open sets $U,V\subset M$ such that $x\in U,\ y\in V \ \text{and} \ \bar{U}\cap\bar{V} = \emptyset$...
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It says to prove that the closed ball in $\mathbb{R}^2$ with the Euclidean metric is this, but isn't that the definition of a closed ball?

Prove that in $\mathbb{R}^2$ $$\overline{B_1((0,0))}=\{x\in \mathbb{R}^2:d_2(x,0) \leq 1\}$$ where $$d_2 (x,y) = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$
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Show that a sequence is Cauchy in $d_1$ iff it's Cauchy in $d_2$

Suppose two distances on $M$ are strongly equivalent, meaning that there exists $k$ and $K$ such that $$d_1(x,y) \le k*d_2(x,y); d_2(x,y) \le K*d_1(x,y)$$.Show that a sequence in $(M, d_1)$ is Cauchy ...
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How does the definition of an open ball in a metric space exclude the empty set?

Definition. Let $(X,d)$ be a metric space. An open ball of radius $r>0$ around a point $x\in X$ is the set $B(x,r) = \{ y \in X \, | \, d(x,y)<r \}$. Looking at the definition of an open ball, ...
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closure of the set $S=\{(x,y)|x\in(0,1)\cap\mathbb{Q};y=\sin(\frac1x)\}$

I am trying to determine the closure of the set $S=\{(x,y)|x\in(0,1)\cap\mathbb{Q};y=\sin(\frac1x)\}$. I don't know where to start. If we define a sequence $(x_n,y_n)=(x_n,\sin(\frac{1}{x_n})),x\in(...
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Prove that the derived set of $A \subseteq \Bbb R$ is closed

For $A \subseteq \Bbb R$, the derived set of $A$, denoted by $A'$, is the set of all limit points of $A$. Theorem: For $A \subseteq \Bbb R$, $A'$ is closed. Does my attempt look fine or contain ...
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1answer
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The image of a continuous mapping on a connected metric space is connected: ($\epsilon - \delta$) proof

I've seen entirely set-theoretical solutions to this proof, but not many based on epsilon-delta arguments. I tried writing one, but am not sure whether my arguments hold. Theorem: The image of a ...
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Distance metric on vector space associated with edges of an undirected graph

Let $G = (V, E)$ be some graph representing for example, a physical road network. Intuitively, I can imagine that I can associate a distance between any two edges $e_1$ and $e_2$, $d(e_1, e_2)$, which ...
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How many of the triangle inequality constraints of a discrete metric space are redundant?

Consider a discrete set of points $X$ and a distance function $d : X \times X \to \mathbb{R}_+$. $d(\cdot,\cdot)$ is said to be a metric over $X$ if the following three constraints are satisfied: $\...
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Metric spaces problem regarding if composition function is uniformly continuous then what about individual function itself

Suppose $X, Y, Z$ are metric spaces and $Y$ is compact. Let $f$ maps $X$ into $Y$. Let $g$ be continuous one-to-one map $Y$ into $Z$ and put $h(x)=g(f(x))$ for $x$ in $X$. If $h(x)$ is uniformly ...
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Topology on the space of functions

Exercise Question ii) I don’t manage to prove that the intersection of open sets is still open. Let’s assume that $U_1,U_2,...,U_m$ are in T. Let $f_0$ be in the intersection of $U_1,U_2, ...,U_m$. ...
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Metric subspace

I have found myself in a tiny problem. If $(X,d)$ is a metric space and $A\subset X $ not empty, then $(A,d´)$ with $d´=d_{AxA}$ is a metric space. Where $d_{AxA}$ is the metric $d$ restricted to ...
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Open/Closed Properties of Word Metric

The Question: Consider the Metric Space $\chi = ${($x_1,x_2,...,x_N$)}, with the word metric d(x,y) = number of digits that differ between x and y. Are the following sets open, closed, or neither? (i)...
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Distance satisfying only positive definiteness

Given a set $X$ and a map $d:X\times X \rightarrow [0,\infty)$. Assume that given $x, y \in X$: $d(x,x) = 0$ $d(x,y) = 0 \Rightarrow x = y$ $d(y,x) = 0 \Rightarrow x = y$ Hence, $d$ is a metric ...
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1answer
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Existence of sequence converging to infimum

Suppose that $f$ is a lower semi-continuous function from a metric space $X$ to $\mathbb{R}$. Does there necessarily exist a sequence $\{x_n\}_{n \in \mathbb{N}}$ in $X$, such that \begin{equation} \...
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Inner product on the Hyperbolic half plane

I think this is the dual question to my previous question. From Gudmundsoon notes, page 60: We can model the hyperbolic space $\mathbb H^m$ as the super half plane space $\mathbb R^+ \times \mathbb ...
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Clarification on Some Notation in a Lemma: Approximating LSC functions by Lipschitz continuous functions

I'm reading a book and it states the following lemma: Let $c \in \mathbb{R}, u: E \rightarrow [c, \infty]$ not identically equal to $\infty$ and define for $t > 0$: $$u_t(x) = \inf \{u(y) + t ...
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1answer
24 views

Step functions and Riemann-Stieltjes Integration

I am reading about Riemann-Stieltjes Integrals in Carother's Real Analysis. I am trying to prove the following Theorem: Theorem: $14.9$ Let $\alpha$ be continuous and increasing. Given $f$ $\...
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1answer
41 views

how we can add a 2-tensor to a 1-form?

In the definition of a Randere norm, we add a Riemannian metric $\alpha$ to a 1-from $\beta$. Indeed, $F(y)=\sqrt{a_{ij}(x)y^iy^j}+b_i(x)y^i$ in ehich $\alpha(y)=\sqrt{a_{ij}(x)y^iy^j}$ is a ...
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Are metrics uniformly equivalent if and only if they have the same zero-distance sets?

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Then $d_1$ and $d_2$ are uniformly equivalent if the identity maps $i:(M,d_1)\rightarrow(M,d_2)$ and $i^{-1}:(M,d_2)\rightarrow(M,d_1)$ are ...