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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Tails sets are Borel

I am trying to proof a particular case of Kolmogorov's law in the set E of infinite binary sequences. Eventually, I'm supposed to prove that a certain type of subsets of this set is in the Borel sigma ...
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6answers
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Rudin's Topological Definition of an Open Set — Does it Disagree with the Metric Space Definition?

I wanted to share this definition of an open set, which made me uncomfortable. It comes from Rudin's Real and Complex Analysis and begins with the definition of a topology: A collection $\tau$ of ...
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6 views

distance metric between multisets

I am trying to define a distance $F(X,Y)$ between two multisets $X$ and $Y$. For each pair of $x \in X , y \in Y$ there exists a distance function $f(x,y)$ which takes the range of $[0,1]$. An ...
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2answers
47 views

Bounded metric on $\mathbb{R}$

I am supposed to give an example of a metric that is bounded on $\mathbb{R}$. In other words, I have to find a function $d:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ which satisfies that $d ...
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1answer
37 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?
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54 views

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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2answers
27 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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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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91 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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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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3answers
79 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
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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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
63 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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3answers
57 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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2answers
80 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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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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3answers
49 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
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
622 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 ...
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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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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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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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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
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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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 ...
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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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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 ...
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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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2answers
137 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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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
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 ...
3
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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 ...
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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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$p$-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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2answers
41 views

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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1answer
22 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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2answers
50 views

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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1answer
41 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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1answer
112 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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1answer
27 views

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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1answer
18 views

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 ...