Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

learn more… | top users | synonyms (1)

2
votes
1answer
33 views

Get a bounded metric from a metric - triangle inequality for $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ [duplicate]

This is related to Proof that every metric space is homeomorphic to a bounded metric space but I can remember that if $d$ is a metric, then $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ is also a metric that ...
1
vote
1answer
16 views

Non-separability of normed spaces

I would like some hints to decide when a normed space is separable or not. I really understood the definition and the classic examples of separable spaces but when I go to show that a space is ...
0
votes
1answer
29 views

To construct a continuous map on compact metric spaces (cf. Gromov-Hausdorff Limit)

This is about Gromov Hausdorff limit on compact metric spaces (Reference A course in metric geometry - Burago Burago and Ivanov, 268p. EXE 7.5.8) Definition : $d_{GH}(X,Y)<\epsilon $ if there ...
2
votes
0answers
27 views

Compactness Theorem (Propositional Logic) and Compactness (Metric spaces). [duplicate]

Definition. A subset $E$ of a metric space $(X,\tau)$ is compact if every open cover of $E$ has a finite subcover. Theorem (Compactness Theorem). A set $\Gamma$ of formulas is ...
1
vote
1answer
23 views

infinite subset of discrete metric space is not compact

The question is Im not really sure how to go about this So far i am trying to show that for an open cover of the infinite subset X, there isn't a finite sub cover and therefore X is not compact I ...
1
vote
1answer
40 views

Metric spaces and normed vector spaces

Studying I learned that there are some theorems and definitions that need a metric structure on the space in which we are working, for example the definition of local maximum needs a metric space or ...
0
votes
3answers
97 views

Why pseudo-Riemannian metric cannot define a topology?

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. Does this imply that in cosmology, say through FLRW metric, ...
0
votes
0answers
14 views

Meaning of amalgamated metric sum of $A_n$’s over $0$ and $d_n$ inherited from $\mathbb{R}^2$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
0
votes
1answer
32 views

What can you say about interior points of a non empty subset of real numbers?

Given that A is a non-empty subset of real numbers, if I(A) denotes the set of interior points of A; then I (A) is:- a) empty. b) singleton. c) a finite set containing more than one element. d) ...
1
vote
2answers
68 views

How calculate with Riemannian metrics (e.g. Multiplication and Divison)?

I have no idea how to handle the following Riemannian metrics, how to find the estimates for the bound and how to actually calculate with $g$ and $d$. Do I need to use the matrix representation? Or ...
1
vote
1answer
15 views

Proving that a set is closed with respect to a defined metric

Let $M = [0,1]^{[0,1]}$ Prove that the set of increasing functions $$ J := \{f \in M : \forall \space a,b \in [0,1], a \leq b : f(b) − f(a) \geq 0 \} $$ is a $d$-closed subset of $M$ where ...
2
votes
0answers
55 views

compact set in a space of functions continuous in $\mathbb{R}$

As it is known the set $\mathcal{B}=\{ f:\mathbb{R}\rightarrow \mathbb{R} : f \mbox{ is continuous}\}$ it is not a metric space with the metric $d(f,g)=sup_{x\in \mathbb{R}}\| f(x)-g(x)\|$ it can ...
3
votes
1answer
49 views

Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space?

Suppose $<M,d>$ is a metric space. Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space? Or maybe lim$_{a \rightarrow b } \space d(a,b) \neq 0 $ implies ...
0
votes
2answers
49 views

Show that mapping is a contraction?

Show that the mapping $f:\Bbb R \to \Bbb R $, $f(x)=1-xe^x$ is a contraction. I tried everything i could think of but i cant get it to work. Witch is not much since i couldn't really find any ...
1
vote
3answers
42 views

Show that the collection of open balls in two metric spaces are identical

I am having trouble trying to prove the following statement. I can see why it would be true intuitively, however, I am having trouble formalising the proof as the notation is quite confusing. Show ...
2
votes
0answers
49 views

Limit of Riemannian manifolds is not Riemannian

I want to prove that $D$, standard unit ball in ${\bf R}^2$ with $|\ |$, with a metric $\| \ \|_1$ is a limit of Riemannian manifolds $X_i$. Here problem is to find $X_i$ (If necessary, all metrics ...
0
votes
0answers
15 views

The name of a polygon defined by multiple overlapping annuli

I am working on a problem in a metric space where points are partitioned into various annuli. If there exists multiple annuli that define a set of points then a polygon can be formed from their ...
1
vote
2answers
48 views

lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$ in Metric Space - Implications

A Metric Space $<M,d>$ is given by the Metric $M$ and distance function $d$ If there exists a Cauchy Sequence $x_n$ such that: lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$, for some $a \in ...
3
votes
1answer
54 views

How to determine the limit of a sequence in a metric space

If I'm trying to prove that a Metric space $(M,d)$ is not complete I have to find a Cauchy a sequence that doesn't converge in $M$. Using the following Metric Space as an example: $M = \{ x \in ...
3
votes
3answers
70 views

In Pugh's analysis book, why are these metric spaces?

Pugh introduces the notion of metric space in chapter 2 as follows Definition: A metric space is a set $X$ equipped with a metric $d$ Clear! For example, a metric space is $(\mathbb{R}, ...
1
vote
4answers
76 views

What does it mean exactly by a metric “generates” a topology?

For example, the discrete metric $d(x,y)$ where $d(x,y) = 1$ if $x\neq y$, $d(x,y) = 0$ if $x = y$ "generates" the discete topology $\tau$ where $\tau = 2^X$ Can someone clarify exactly what is meant ...
3
votes
1answer
63 views

On preimage of open sets of functions on real line having at most countably many discontinuity points

Let $f:\mathbb R \to \mathbb R$ be a function whose set of discontinuity points is at most countable ; is it true that for every open set $G \subseteq \mathbb R$ , there is an open set $U$ and a ...
2
votes
1answer
42 views

Showing that the following two metric are equivalent.

Let $(X,d)$ be a compact metric space, and $f \colon (X,d)\to \mathbb{R}$ is a continuous function such that if $x,y \in X$ and $x \neq y$ than $f(x) \neq f(y)$. Let $t \colon X \times X \to ...
2
votes
1answer
70 views

what are the geodesics in the hyperbolic upper half plane?

In the upper half-plane $$ H = \{(x, y) \in \mathbb{R}^2 \mid y > 0\} $$ the distance between the two points (a,A) and (b,B) is set by the shortest curvature in metrix $$ F(y) = \int_a^b ...
-1
votes
1answer
49 views

Find a metric $d$ on $X$ such that $(X, τ^{(d/X)})$ is not connected

$X = (\{0\} \times [-1,1] \cup \{(x,\sin(π/x)) : x \in (0,1]\} \subset \mathbb{R}^2$ Find a metric $d$ on $X$ such that $(X, τ^{(d/X)})$ is not connected. Note: $τ^{(d/X)}$ denotes the metric ...
1
vote
0answers
85 views

Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in ...
0
votes
1answer
19 views

Metric space with two similar points which are not in the same orbit.

Is there an example of a metric space $X$ with two points $p$ and $q$ so that for every $r>0$ the ball with radius $r$ and center $p$ is isometric to the ball with radius $r$ and center $q$ and yet ...
0
votes
2answers
51 views

How can I prove that

Let we have the following ultrametric space $(z,|.|_2)$ such that if $x=r.2^n$ then $|x|_2=2^{-n}$ how can I prove that the topology produced by this metric isn't discrete topology ?
0
votes
0answers
55 views

If we think of infinity as a number, how does it affect the compactness/completeness of a metric space?

I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. ...
3
votes
2answers
58 views

A metric on the natural numbers

Does there exist a complete metric on the set of natural numbers such that $\{n,\,n+1,\,n+2,\,\cdots\}$ is a closed ball for each $n$?
0
votes
0answers
51 views

Mahalanobis distance

Suppose there is a function $f$, for which we know the inequality $$f(r)\leq r$$ is true, where $r=||x-y||_2=\sqrt{(x-y)^T (x-y)}$ is the Euclidean distance. If now we use the Mahalanobis distance ...
2
votes
1answer
27 views

Verification of proof of continuity between metric spaces and deduction from proof

Let $M = [0,1]^{[0,1]}$ and $d(f,g) = \sup{\{\lvert f(x) - g(x)\rvert \mid x \in [0,1]\}}$. For $a,b \in [0,1]$ let $\phi_{a,b}(f) = f(b) - f(a)$ ($\phi$ maps from $M \to \Bbb{R}$). Assume that ...
1
vote
2answers
33 views

compact metric spaces and infimum

I am currently revising metric spaces and have come across a question which I am unable to answer and have no idea how to begin with. Let $(M,d)$ be a compact metric space. Suppose $T \colon M \to ...
4
votes
1answer
67 views

General approach to determine completeness of metric space

I've looked at a few questions online asking to determine the completeness of Metric Spaces. 2 such examples of metric spaces $(M,d)$: 1) $M = \{ (x,y) \in \mathbb{R}^2 \space : y>0 $ or $ x=0=y ...
2
votes
5answers
74 views

For a nonempty subset $A$ of a metric space, $x \in \overline{A}$ iff $d(x,A) = 0$.

Let $(X, \rho)$ be a metric space and $x \in X, A \subset X$ ($A \neq \varnothing$). Then $x \in \overline{A}$ iff $d(x,A)=0$. I am facing difficulties showing that $d(x,A)=0$ implies that $x ...
4
votes
3answers
439 views

Does a connected countable metric space exist?

I'm wondering if a connected countable metric space exists. My intuition is telling me no. For a space to be connected it must not be the union of 2 or more open disjoint sets. For a set ...
0
votes
1answer
16 views

contraction mapping unique solution

I am currently revising metric spaces and have come across a question slightly different to ones i am used to and would appreciate any hints as to how to attempt this question here is a photo of the ...
1
vote
2answers
53 views

Why don't clopen sets pose problems in the “preimage” definition of continuity, and in the definition of a topology?

Recently I have been wrestling with the reasoning behind why clopen sets do not lead to contradictions when we define continuity in terms of open/closed sets, the topology of metric spaces, and other ...
1
vote
0answers
29 views

Given $2$ closed subsets whose union and intersection are path connected. Show that each subset is path connected.

Let $A$ and $B$ be $2$ closed subsets of a metric space $E$. Suppose $A \cap B$ and $A \cup B$ are path connected. Show that $A$ and $B$ are path connected. Proof: it suffices to show that A is path ...
2
votes
2answers
33 views

complete spaces and the baire principle

There is this problem that I can't get an answer to. Begin with $(X,p)$ a complete metric space, with no isolated points. (This means that every point is an accumulation point, right?) Now prove ...
1
vote
1answer
38 views

Baire principle with open and dense subsets-edited

This is a question on the Baire principle for metric spaces. Let $X$ be a COMPLETE metric space without isolated points. Prove or disprove that, every sequence $(O_n)$ of open and dense subsets of X ...
0
votes
0answers
24 views

Is there a minus thickening operator on a metric space?

Let $S$ be a metric space and $A$ a subset. For some $\varepsilon>0$ define the $\varepsilon$-thickening of $A$ as $$A^{\varepsilon} = \left\{p \in S \mid \exists q \in A \;\;\text{with}\;\; ...
0
votes
0answers
30 views

Exercise from *Apostol's Mathematical Analysis*:

Exercise from Apostol's Mathematical Analysis: A point $x\in \Bbb R^n$ is a condensation point if every ball has the property that $B(x)\cap S $ is uncountable. Assume that $S\subset \Bbb R^n$ and ...
1
vote
0answers
43 views

Subspace or not?

I'm studying Functional Analysis and I got a doubt about the following theorem: Let $H$ be a prehilbert space and $S \subset H$ complete and convex. Then, $\forall x \in H$ there exists a unique ...
3
votes
2answers
139 views

Prove wrong the following statement about metric spaces and completeness

Statement: Given the condition: $d(x,y)^2 \leq g(x,y) \leq d(x,y) \space \forall \space x,y \in M$ If $(M,d)$ is complete then $(M,g)$ is complete Question: Prove or provide a counterexample to ...
2
votes
2answers
38 views

Isosceles Triangles in Hilbert Spaces and Metric Spaces generally

In what types of metric spaces $\langle X, d \rangle$ is it possible to do the following? Task: For any two points $x, y \in X$ such that $d(x,y) \leq 2\epsilon$, find a third point $z$ such that ...
3
votes
1answer
86 views

Determining compactness and completeness of metric space

Metric Space: (M,d) Set: $M = \{ (x,y)\in \mathbb{R}^2:y>0 $ or $ x=0=y \}$ Metric: $d((x,y),(a,b)) = $min$\{ $max$ \{ |x-a|,|y-b| \},y+b \}$ $\space$ completeness: $\lim_{n ...
1
vote
1answer
19 views

Can there be metrics on sets of random variables?

First off - I do not know much probability theory, so please pardon me if this question is nonsensical. The question arose from the following thought: can I make the expectation function continuous, ...
0
votes
1answer
32 views

Properties of oscillation

Let $(X,\tau)$ be a topological space, $(M,\operatorname{d})$ a metric space and $f:X \to M$. Then Definition: The oscillation of $f$ at $x \in X$ is $$ \omega_f(x) = \inf_{x \in U \in ...
1
vote
2answers
52 views

Difference between completeness and compactness

According to Wikipedia: A metric space $M$ is said to be complete if every Cauchy sequence converges in $M$ $ $ A metric space $M$ is compact if every sequence in $M$ has a subsequence ...