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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How to make the Symmetric Distance a metric?

I am trying to construct a family $S$ of measureable subsets or $R^2$, on which the symmetric difference, defined as: $SD(A,B) = Area(A\setminus B \cup B \setminus A)$, is a metric, i.e., different ...
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1answer
35 views

metric spaces and topology [on hold]

Let $d_1,d_2$ be metrics on $X$ such that any sequence $(x_n)$ converges in $(X,d_1)$ iff it converges in $(X,d_2)$ to the same point. Must $(X,d_1)$ and $(X,d_2)$ have the same topology?
4
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2answers
87 views

Topological spaces vs. metric spaces

Are there any "realistic" examples of topological spaces that are not metric spaces. You are free to invent your own definition of "realistic". But, at a minimum, a realistic example is one that ...
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73 views
+50

Continuity of Function Related to $F$-norms

Let $X$ be a locally bounded $F$-space and $\left\|\cdot\right\|$ be an $F$-norm on $X$. Suppose that $\left\|\cdot\right\|$ is concave: for all $x\in X$ fixed, the function ...
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2answers
69 views

Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
4
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3answers
78 views

When a group acts on a metric space isometrically, transitively, and faithfully, is it a closed subspace of the isometry group?

Suppose you have a group $G$ acting on $ (M,d)$ a compact metric space by isometries (meaning $d(gx,gy) = d(x,y)$ for all $x,y \in M$ and all $g \in G$), transitively and faithfully. You can define ...
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1answer
140 views

Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line? [closed]

Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Is there a theorem that describes the form of the elements of $X$? Context For open subsets of the line, such a ...
19
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4answers
2k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
2
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2answers
324 views

A question on countability of isolated points of a subset of R

The question is to prove that with respect to the euclidean metric on the Real numbers prove that if A is any subset of R, isoA is countable and hence deduce that if A is uncountable the A' is ...
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1answer
27 views

Two definitions of compact set

I am reading parallely two books on analysis, and they have two different definitions of compact set: 1) Subset A of metric space X is called compact, if every open cover of A contains a finite ...
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2answers
26 views

Different metric structure on $\mathbb{Z}$

Is it possible to equip $\mathbb{Z}$ with a metric such that the closed sets are precisely the finite subsets and $\mathbb{Z}$?
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47 views

Metrizability of quotient spaces of metric spaces

Suppose $X$ a metric space and $\sim$ an equivalence relation on $X$. Is the space $X/\mathord{\sim}$ metrizable? I think that the answer is no, but I could not arrive at a counterexample.
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1answer
36 views

Connected subsets of metric (or T1) spaces

I have proved some statements about connected subsets of a metric space. They are really basic, but I want to make sure that they are true. Would someone please tell me whether these statements are ...
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3answers
34 views

Proving that a function on a compact metric space is bounded above

The question is as follows. Let $(X,d)$ be a compact metric space, and let $f:X \rightarrow \mathbb R $. Assuming that for each $r \in \mathbb R$, the set $G_r=\{x \in X : f(x) \lt r\}$ is open, prove ...
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0answers
14 views

Does a strictly convex and weak metrizable unit sphere of a Banach space imply separability?

I want know If $X$ be a Banach space with strictly convex norm and a metrizable unit sphere, $S_X=\{x\in X: \|x\|=1\}$, for the weak topology. Does a strictly convex and weak metrizable unit ...
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2answers
61 views

If $A$ and $B$ are closed subsets of the set of real numbers, then is $A+B$ closed? [duplicate]

Let $A$ and $B$ be two closed subsets of the set of real numbers. Define $A+B=\{a+b\in\mathbb{R}:a\in A ,b\in B\}$. Is it true that $A+B$ is closed in $\mathbb{R}$? If not, could you give a ...
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2answers
361 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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1answer
231 views

Non-empty intersection of open balls in $R^n$ contain open balls

I want to prove that if the intersection of two open balls about the points $x, y$ (resp.) is non-empty, then there exists a third ball centered at some point $z\in B_{\epsilon 1}(x)\cap B_{\epsilon ...
2
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3answers
55 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
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1answer
16 views

For every real number there are exactly two isometries of the real line that leave it fixed

I have some preliminary questions before I attempt this problem in my book. If $M$ is the metric space of all the real numbers and $x_0 \in M$, prove that there exist exactly two isometries of $M$ ...
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1answer
179 views

Isoperimetric inequalities of a group

How do you transform isoperimetric inequalities of a group to the of Riemann integrals of functions of the form $f\colon \mathbb{R}\rightarrow G$ where $G$ is a metric group so that being ...
2
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1answer
22 views

Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
2
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1answer
22 views

Equivalent Metric on Finite Set

Suppose $(X, d)$ be a finite metric space. I agree that all the metrics on finite set X are equivalent. Can any one prescribe the methodology to derive all equivalent metric to the metric $d$? Given ...
4
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2answers
621 views

Can the distance between 2 non-empty sets be infinite?

Intuitively I would immediately assume no, but that's not how things usually work in math and considering there are different kinds of infinities I haven't been able to find the answer. Here's my ...
2
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1answer
36 views

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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136 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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58 views

$ d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}? $

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$ \quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}} $$ ...
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2answers
30 views

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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1answer
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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1answer
54 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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2answers
35 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 ...
2
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2answers
49 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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1answer
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Is there a metric $d$ such that it makes a countably infinite set $X$ not a discrete metric space $(X,d)$? [closed]

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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1answer
95 views

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 ...
3
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1answer
1k views

Every subsequence of $x_n$ has a further subsequence which converges to $x$.Then the sequence $x_n$ converges to $x$.

Is the following is true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. I ...
3
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0answers
52 views

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 ...
0
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1answer
26 views

Finsler Metric from page 2 of the book by Chern and Shen.

Physicist here not a mathematician. I am trying to understand the notation for the Finsler metric in Chern and Shen's book. The equation is $$\textbf{g}_y(u,v):=\frac{1}{2} ...
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1answer
40 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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0answers
29 views

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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1answer
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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1answer
35 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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1answer
65 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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1answer
17 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. ...
2
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1answer
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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1answer
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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2answers
67 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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1answer
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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0answers
97 views

lebesgue measure is metric outer measure

This question is driving me crazy. I need to prove that Lebesgue measure is metric outer measure. Unfortunately, I get lost. All I have is because $m$ is Lebesgue measure, $m^*(A \cup B) < ...
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2answers
49 views

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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3answers
34 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 ...