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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Metric on the Set of Binary rectangular matrices

Consider a set of all possible Binary rectangular matrices. How many non-equivalent metrics can be defined? How to define non equivalent metrics on this set precisely?
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Proving no finite basis of the system of neighborhoods at $a$ in the real line exist.

I'm not sure how to prove it, the gist is: I need to find the "smallest" neighborhood in the basis, take a ball of half that radius and show "look, there is no member of the basis in this ball, thus ...
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7 views

Projection from pseudometric space into metric identification preserves topologies

I've just started working through a topology textbook, and I'm sure I'm being silly, but I can't for the life of me make any headway on the following question: Let $(M,\rho)$ be a pseudometric space, ...
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50 views

What is the difference between $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ in terms of their metrics?

what is the difference between $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{Q}$ in terms of their metics? Do I need more assumptions to make difference between them beside just their metric functions?
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55 views

Is there a metric $d$ such that it makes a countably infinite set $X$ not a discrete metric space $(X,d)$? [on hold]

Let $X$ be a countably infinite set. Then, is there a metric $d$ such that $(X,d)$ is not a discrete metric space?
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48 views

Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
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37 views

Show that for $(X,d)$ a metric space, $U= \{x \in X: d(x, C) \leq d(p, C)\}$ is a closed set

Let $(X,d)$ be a metric space, $C$ be a closed set in $X$. Define $$d(C, x) := \inf \{d(c, x): c\in C \}$$ for all $x \in X$. Fix a point $p \in X$. Show $U= \{x \in X: d(x, C)\leq d(p, C)\}$ is a ...
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Inequality proof using the triangle inequality

I am reading Kreyszig's Intro to Functional Analysis and am a bit stoked with one of the problems (problem 12 in section 1.1, page 9): Problem: Given a metric space $(X, d)$, show, using the ...
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34 views

Is there a metric space of an infinite set such that every closed set is finite except the whole space.

Let $X$ be an infinite set. Then, is it possible to construct a metric space $(X,d)$ such that every closed set except the whole space $X$ is finite? If possible, what would be the example of such ...
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Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
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Number of subsets/open subsets/closed subsets of a metric space.

Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces which have the same infinite set $X$, but the different metrics $d_1$ and $d_2$. Denote the collection of subsets $X$ by $S$, and the collection of ...
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25 views

Confused on the definitions of norm of a function.

If $$\|f \| =\sup \{|f(x)|:x \in [0,1]\} $$ and also $$ \|f \|=\int^1_0 |f(x)| \, dx,$$ then for $f(x)=x$, we have $\sup \{|f(x)|:x \in [0,1]\} = 1$. But $\int^1_0 |f (x)| \, dx = \int^1_0 |x| \, ...
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32 views

What is the difference between an isometric operator and a unitary operator on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is the identity operator) What is the difference between ...
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62 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ of ...
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16 views

Is this theorem about “completion of metric space” correct?

It's well-known that there is a completion of a metric space unique upto isometry. I have tried to modify this theorem slightly and I proved this statement: Let $(X,d_X)$ be a metric space. ...
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29 views

Cauchy sequences are bounded in every metric space

A few days laid out an example, and asked for help, and @ shadow10 replied, his answer the question of can I find the Every Cauchy sequence is bounded but please someone help me in relation to ...
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110 views

Two distinct geodesics joining two points on a compact manifold

This is a problem from the book Gallot, Hulin, Lafontaine: Riemannian geometry (3rd edition). Exercise 2.118: For a compact Riemannian manifold, let $p,q$ two points such that $d(p,q) = ...
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65 views

If a property holds for arbitrary compact set in a metric space, does it also holds for the metric space?

Suppose a metric space $(X, d).$ Further suppose that a property $A$ holds for arbitrary compact subset of $X.$ Does the property $A$ also hold for $X$? Context I hoped for some general theorems of ...
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30 views

Definition of a metric-nonnegativity condition

There is a question in my mind which seems to be silly but I am desperately wanting the answer. Why a metric is defined from $X\times X$ to $\mathbb R$ and not to the set of nonnegative reals? I ...
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Why the continuity of a function on a metric space doesn't depend on metrics?

In the definition of the continuous function on a metric space, it seems to me that a continuous function depends on the metric of the given metric space. Could somebody explain Why the continuity of ...
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33 views

What does it mean by that two different metrics may define the same collection of open sets?

What does it mean that two different metrics may define the same collection of open sets? The assumption is that a given set is equipped with two different metrics to form two different metric ...
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22 views

Differentiability of rectifiable curves

I have the following question. Let $\gamma:[a,b]\rightarrow X$ be a rectifiable curve in a metric space $(X,d)$. If we consider the length function of $\gamma$, $L:[a,b]\rightarrow [0,L(\gamma)]$, we ...
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$p$-adic metric proof

I need to prove this, Let $p$ be an odd number. It is defined the function $v_p:\mathbb{Q}\to \mathbb{Z}$ as $$v_p\left(p^n\frac{a}{b}\right)=n, \hbox{ if } \mathrm{mcd}(a,p)=\mathrm{mcd}(b,p)=1.$$ ...
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Existence of a metric space M with no continuous map from M to any other metric space

Is it possible to have a metric space M such that there is no continuous map from M to any other metric space?
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21 views

Unit ball of continuous functions is a closed set - Proof with neighborhood argument

This question is trivial if one uses sequence definition, but I want to use the usual topological definition of closed set. That is , a set is closed if its complement is open. Let $U=\{f\in ...
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Looking for a proof that the diameter of the smallest bounding circle is less than or equal to $\frac{2}{\sqrt{3}}$ times the diameter of the set

This came up while I was attempting to solve an old journal problem. It's not the easiest result to search for so I figured I would ask. Let $E$ be a subset of $\mathbb{R}^2$, then the diameter of ...
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39 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
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111 views

What is an ultrametric normed vector space?

Wikipedia's article on ultrametric spaces seems to suggest that an ultrametic space can also be a normed vector space. It seems to be impossible for an ultrametric to be induced by a vector space ...
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Metric for vector sets

I am currently working on a classification algorithm. Each class is represented by a set of 3D vectors. The cardinality differs for each class. The order of the vectors in a set is completly random. ...
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Definition of a metric space with bounded growth

Does anyone know the definition of a metric space with bounded growth? I was reading a paper by Roe titled Hyperbolic groups have finite asymptotic dimension, where he writes a definition, but I ...
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In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
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Lebesgue prolongation of a measure and metric spaces

Anyone knows the connection between these two things. Tha books talks about the measure $m^*$ in a ring $\mathcal{R(G_m)}$; $m^*(A \triangle B)$ can be seen like a distance between two sets of the ...
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Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $X$ be a metric ...
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Cauchy sequence in compact metric space

Suppose $f:X\rightarrow X$ continuous function, $X$ is compact metric space with $\rho(f(x),f(y))<\rho(x,y)$ for any $x\neq y$. Let $x_n=f(x_{n-1})$, with $x_0\in X$ arbitrary. I want to show that ...
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Does “uniformly isolated” imply closed?

Let $X$ denote a complete metric space and consider a subset $A \subseteq X$. Call $A$ uniformly isolated iff there exists $r > 0$ such that for all $a \in A$, we have that $B_r(a) \cap A = \{a\}$. ...
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amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
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Homeomorphism of Interior of Convex Polygon to Open Unit Disk

Show that the interior of a non-degenerate convex polygon in $\mathbb{R}^2$ is homeomorphic to the open unit disk in $\mathbb{R}^2$. My attempt: Let $P$ be the set of points in the polygon, let ...
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Two forms of application of metric tensor to get differential length

I'm reading the monograph "Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity" by Joseph Kolecki, now retired, of NASA. I have a ...
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67 views

Is there any proof for this simple observation? [duplicate]

If we consider the Euclidean space $R^3$, it is simply the space where we live. Here we can find only four point such that distance between any two points is a constant. If we consider the Euclidean ...
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54 views

Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
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Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
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Discontinuous function whose restriction on closed sets is continuous

Let $X$ a metric space, $\{U_i\}$ a collection of non-empty closed sets whose union is all of $X$. Give an example of a function $f:X\rightarrow \mathbb{R}$ such that the restriction $f|_{U_i}$ is ...
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Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
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Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
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If $S \subseteq X$ is closed, is $f(S,r)$ necessarily closed?

Let $X$ denote a metric space. Whenever $S \subseteq X$ and $r \in \mathbb{R}_{\geq 0}$, write $f(S,r)$ for the following set. $$\{x \in X \mid \exists s \in S : d(x,s) \leq r\}$$ Question. ...
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42 views

Gromov hyperbolic metric spaces are quasi-convex

I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a ...
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1answer
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Find continuous $f$ s.t. $f(x,y)^2 = x^2$ ($f: H \to \mathbb{R}$, H connected)

(i) Show that a metric space M is connected if and only if every continuous integer-valued function on M is constant. (ii) Show that $H = \{(x, y) \in R^2 : x > 0\}$ is connected. By ...
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45 views

Proving there exist convergent subsequences for bounded sequence of real numbers

I'm trying to teach myself some basic topology by self-studying from Intro to topology by Mendelson. I'm stuck on one of the exercises and can't figure out how to proceed with the proof. The question ...
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33 views

Prove that square metric on $\mathbb{R}$ is in fact a metric.

In Munkres's topology, he proves that square metric on $\mathbb{R}^n$ is in fact a metric. By the square metric, I mean this function: $P:\mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{R}$ ...
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Easier proof of “countable hypocompactness”

I am interested in the following result, which appears as an old qual problem: Let $X$ be a metric space and $\{U_i\}$ a countable open cover. Prove that there exists a countable open refinement ...