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.

learn more… | top users | synonyms (1)

4
votes
0answers
66 views

Question on complete metric spaces and whether the following is a complete metric space:

Let $ S \subset C^2([0,1])$(set of all two-times differentiable functions on $[0,1]$), which satisfy $$f(0)+f(\frac{1}{2})+f(1)=0.$$ Question :Is $ (S,d)$ is a complete metric space, where $d$ is ...
0
votes
0answers
33 views

Topology Question in Munkres Analysis on Manifolds

Let X be a metric space with metric d. Let Y $\subset$ X. Give an example where A is open in Y but not open in X. Give an example where A is closed in Y but not closed in X. I'm stuck on question ...
0
votes
2answers
25 views

What is the difference between 1) given any , 2 ) for every 3) for all in definitions?

I have a lot of confusion. Definition: A sequence $\{x_n\}$ of points in a metric space is said to converge if there is a point $p \in S$ with the following property: For every ...
-1
votes
1answer
57 views

Prove $d(x,y)=(x-y)²$ is a metric

How can I prove that $d(x,y)=(x-y)²$ is a metric? I have proven all properties except the triangle inequality, i.e $d(x,y) \le d(x,z)+d(z,y)$ I have tried to prove it using the fact that ...
5
votes
1answer
121 views

Metric limit and limit in category

Is it possible to construct a category $\mathcal{C}$ with $\mathrm{Ob}\,\mathcal{C}=\mathbb{R}$ and for every diagram of the from $$a_0\leftarrow a_1\leftarrow\cdots a_n\leftarrow\cdots$$ the inverse ...
6
votes
1answer
32 views

A proper subspace of a normed vector space has empty interior.

In a vector normed space $E$, prove that all vectorial subspace $F\neq E$ has a interior empty. My approach:We consider, the open ball $B\subset F$, with $F$ proper subspace of $E$. If $x\notin F$, ...
5
votes
1answer
57 views

Continuity of a “minimal distance” projection $f:(X,d) \to (K, d_{|K})$ for a compact $K \subset X$. (Hint preferred)

Let $(X,d)$ be a metric space and $K$ be a compact subset of $X$. Show that for every $x \in X$ there exists $k_x \in K$ such that $$d(x,K)=d(x,k_x)$$ Suppose that for every $x\in X$, there exists ...
1
vote
1answer
53 views

Countable choice and totally bounded metric spaces

Can we prove that the following statement is equivalent to the axiom of countable choice (CC)? If every sequence in a metric space $X$ has a Cauchy subsequence, then $X$ is totally bounded. ...
0
votes
0answers
47 views

Open Set in the Cartesian plane.

I'm trying to prove that the following set is an open set in $\mathbb{R}^2:$ $$A=\{(x_{1},x_{2})\in\mathbb{R}^{2}: x_{1}+x_{2}>1\}$$ with respect to norm $||x||_{1},||x||_{2},||x||_{\infty}.$ ...
2
votes
1answer
36 views

Image of isometric immersion

Let $M$ a metric space with the following property: For all isometric immersion $f:M\to N$, the image of $f(M)$ is a open set in $N$. Prove that $M$ is empty set. A function $f:M\to N$, is called ...
1
vote
0answers
22 views

Find the $\epsilon - \delta$ values for the continuous function - modified step function defined on $[0,1]$

Let the modified step function be defined on $[0,1]$ by : $f(x) = \begin{cases} \bigg( \dfrac {2^n+1}{2}\bigg )x - \dfrac {2^n-1}{2^n} ; & n \in \mathbb N~~ , \dfrac {2} {2^n+1} ...
0
votes
0answers
14 views

Geers MPC metric - understanding magnitude error and phase error

It is regarding this question/subject Integrate two sets of Data and check similarity I'm applying the formulas mentioned [here]. But I still can't understand these error rates M and P. It is ...
2
votes
1answer
45 views

Prove the following theorem involving homeomorphic metric spaces.

I want to prove this theorem: If $(X,d)$ and $(X,d')$ be homeomorphic metric spaces, then they have the same convergence sequences. However, there exists homeomorphic metric spaces $(X,d), (X,d')$ ...
0
votes
1answer
26 views

Completion of metric-space, when “corresponding” Euclidean space is complete

I somehow couldn't find the answers to several probably simple questions. I am new to the topic, thus, please excuse any lack of knowledge. Let $A_1\subseteq\mathbb{R}^n$, such that the metric-space ...
0
votes
2answers
38 views

Let $f$ be continuous on $M=A\cup B$, then $f$ is continuous on every $x\in A\cap B$.

Let $M=A\cup B$, a metric space. If $f:M\to N$ is such that $f|A$ and $f|B$ are continuous, then $f$ is continuous in each point $x\in A\cap B$. My approach: If $f:M\to N$, is such that $f|A$ is ...
1
vote
1answer
29 views

How to define a “line” and “symmetry w.r.t. a line” in $L_2(\lambda)$ space

For any $x,y\in L_2([0,1],\lambda)$, define the inner product $\langle. , . \rangle$ by \begin{equation} \langle x, y \rangle=\int_{[0,1]} x(t) y(t) \lambda (dt) \end{equation} Is it proper to ...
0
votes
1answer
24 views

Equivalence between metrics, show that $d_1\sim d_2\sim d_3$.

Let $d_1,d_2$ and $d_3$ metrics on the metric space $M$. If $d_3\succ d_2\succ d_1$ and $d_1\sim d_3$, then $d_1\sim d_2\sim d_3$. Edit: Where $d_1\succ d_2$, mean that $d_1$ is more fine that $d_2$, ...
-2
votes
3answers
53 views

Subsets of a metric space

Let $U$ be an open subset of a metric space $X$, and let $A \subseteq X$. Show that $U \cap A = \emptyset$ implies $U \cap \overline{A} = \emptyset$. I think that, since $U$ is an open subset of ...
6
votes
1answer
72 views

$R^2$ is not isometric to $R^3$

Is there a direct proof for showing that $R^2$ is not isometric to $R^3$ (with the usual metrics)? I know that they are not homeomorohic but I think there should be some direct and easy proof for ...
3
votes
3answers
81 views

Extension of metric definition to two sets

The standard definition of a metric, is a function $d: X\times X \to \mathbb{R}$. What is a sensible/common extension of a metric/pseudometric to $\tilde d: X\times Y \to \mathbb{R}$, i.e. distance ...
0
votes
3answers
44 views

Show that exist open ball B, such that $f(B)\cap g(B)=\emptyset$

$M,N$ metric space. Let $f,g:M\to N$ continuous in a point $a\in M$. If $f(a)\neq g(a)$, then exist a open ball B of center a, such that $f(B)\cap g(B)=\emptyset$. In particular, if $x\in B$, then ...
1
vote
1answer
33 views

Name for a 'perimeter' metric?

Suppose we have a space like $\partial([0,1]^2)$, the boundary of the unit box, simple a square. I'm looking for a metric that would measure the distances that must be traversed to get from one point ...
0
votes
0answers
59 views

Metric-space complete?

My question is if a specific metric-space is complete, respectively under which conditions it is complete. I am rather a newby, but hope that the question is understandable. The metric-space is ...
1
vote
1answer
32 views

Why does the Hausdorff metric need to be defined on bounded subsets only?

Claim: Suppose $X$ is a non-empty set and $d$ is a metric on $X$. Let $S(X)$ denote the collection of all non-empty closed bounded subsets of $X$. For each $A$ and $B$ in $S(X)$, define $$h(A,B) = ...
2
votes
2answers
19 views

A k-lipschitz function

Let $f:M\to \mathbb{R}$ a k-lipschitz function, i.e, $\vert f(x)-f(y)\vert\leq kd(x,y)$, for any $x,y\in M$. Show that $f(x)=\displaystyle \inf_{y\in M}[f(y)+kd(x,y)]=\displaystyle\sup_{y\in M}[f(y)-k ...
2
votes
2answers
42 views

Open balls with radis $>\epsilon$ in a compact metric space

In a compact metric space $(X,d)$, for a given $\epsilon>0$, if $(x_j)_{j \in J}$ is a family of points of $X$ such that the balls $B(x_j, \epsilon)$ are pairwise disjoint, does it automatically ...
0
votes
0answers
9 views

Show that the sphere of center $(a,b)$ is equal to $(B[a,r]\times S(b,r))\cup (S(a,r)\times B[b,r])$

Let $d[(x,y),(x',y')]=max\{d(x,x'),d(y,y')\}$, show that the sphere of center $(a,b)$ and radius $r$ in $M\times N$ is equal to $(B[a,r]\times S(b,r))\cup (S(a,r)\times B[b,r])$. I know that the ...
18
votes
5answers
1k views

Counterexample to “ a closed ball in M is a closed subset.”

I am studying topology, on my own, using a text I found online. I am currently reviewing the “Metrics” section that reminds me of the real analysis course I took over 10 years ago. The text ask me to ...
1
vote
4answers
82 views

Show that the rationals are an incomplete metric space without reference to reals

I know that you can create rational sequences that converge to irrationals, but is there a simple way to do this without explicit assumption of the existence of the reals? I'm thinking of something ...
2
votes
2answers
37 views

Normed vector space, show that $d(a,c)=d(a,b)+d(b,c)$.

In a normed vector space E, if $c-a=t(b-a)$, with $t\geq 1$, then $d(a,c)=d(a,b)+d(b,c)$. I see this problem like, a vector with extremes , a and c, then $c-a=t(b-a)$ is a separation of the segment ...
0
votes
2answers
18 views

Question on p. 11 of Bryant, Metric Spaces book

On p. 11, the last paragraph says: Confirm by direct integration that $x(t)=1/(1+t)$ is indeed a root of the equation $$ x(t)=1-\int_0^t [x(u)]^2 \mathrm{d}u. $$ I am a little confused on how to carry ...
0
votes
1answer
12 views

Range of a function taking matrices to matrices contains a given set

Consider $M_2(\mathbb{R})$ to be the space of all real $2\times 2$ matrices. Define $$S_r:=\{A\in M_2(\mathbb{R})\lvert \max_{i,j}|a_{ij}|\leq r\}$$ for $r>0$. Now let $\Psi :M_2(\mathbb{R})\to ...
5
votes
5answers
151 views

An Example of a Nested Decreasing Sequence of Bounded Closed Sets with Empty Intersection

Could someone provide me with an example of a metric space having a nested decreasing sequence of bounded closed sets with empty intersection? I first thought of Cantor set but the intersection is not ...
0
votes
1answer
24 views

If $X$ admits a nearest point to each point in every metric super space of $X$: every point like function on $X$ attains it's minimum value on $X$.

If $X$ admits a nearest point to each point in every metric super space of $X$, then every point like function on $X$ attains it's minimum value on $X$. Definitions used: Suppose $(X, d$) is a ...
0
votes
1answer
23 views

Does totally bounded imply covering compact?

Let S be totally bounded. So, $\forall\epsilon>0$, $S$ can be covered by a finite number of balls of radius $\epsilon$. Now, let $\{S_n\}$ be a cover of $S$ with open sets. So, $S_n\cap S$ is also ...
2
votes
1answer
47 views

The assumption of nonemptiness in the theorem (3.10 from Rudin) about the intersection of nested compact sets

There is Theorem 3.10(b) in baby Rudin. If $K_n$ is a sequence of compact sets in a metric space $X$ such that $K_n\supset K_{n+1}$ and if $$\lim_{n\to \infty}\text{diam}K_n=0,$$ then ...
2
votes
0answers
60 views

Are local quasi-geodesics already quasi-geodesics in hyperbolic spaces?

Recall the following definitions 1) A $(\lambda, \varepsilon)$-quasi-isometric embedding $f$ between metric spaces $X$ and $Y$ is a map $X \to Y$ such that $\frac{1}{\lambda} d_X(x,y) - \varepsilon ...
1
vote
1answer
68 views

Prove set $\left\{(x,y)\in\mathbb R^{2} : \frac{x^2}{4}+\frac{y^2}{9}<1\right\}$ is open

How to show prove that the set $\left\{(x,y)\in\mathbb R^{2} : \frac{x^2}{4}+\frac{y^2}{9}<1\right\}$ is a open set? my attempt: Graphically it being a interior of ellipse, without boundary ...
3
votes
4answers
69 views

Show the set $\{x \in \Bbb Q : x>0, 2<x^2<3\}$ is open and closed

Show that in the metric space $\Bbb Q$, with the usual metric (given by the absolute value), the set $$S=\{x \in\Bbb Q : x>0, 2<x^2<3\}$$ is both open and closed. My attempt: I need to ...
0
votes
0answers
22 views

Find the cluster points of a set

Given the set $A=[a,b)$, with $a,b \in \mathbb{R}$ and $a< b$, write the set of ALL cluster points of A. Okay so i understand the definition of a cluster point, knowing that for any ...
0
votes
0answers
16 views

every infinite bounded subset of X has an limit point in X and CC

Related to questions and definitions used in question: Nearest point property and WeistrassB copactness of each infinite bounded set Proof of one side of theorem: every bounded infinite bounded ...
0
votes
0answers
13 views

Nearest point property and WeistrassB copactness of each infinite bounded set

Definition: Let S subset of X where (X, d) is metric space. Then S is said to have nearest point property to x in X iff dist(S, x) = d(s,x) for some s in S. Theorem: A metric space X admits a nearest ...
10
votes
1answer
123 views

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
0
votes
1answer
21 views

$X$ be a complete metric space and $f:X \to X$ be a bijective and connected preserving map ; then is $f^{-1}$ also connected preserving?

Let $X$ be a complete metric space and $f:X \to X$ be bijective and a connected preserving map i.e. $f$ carries every connected set of $X$ to a connected set of $X$ ; then is it necessarily true that ...
4
votes
2answers
63 views

Is every metric space quasi-isometric to a graph?

I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$. During this proof I've precisely used the fact that given two point in $X$ there ...
2
votes
1answer
47 views

Understanding convergence in normed spaces and the language used when talking about norms.

We have the following definition about convergence in a normed space: "Let $(x_n)_{n=1}^\infty$ be a sequence in a normed space $(X,\|\cdot\|)$. We say that $x_n\to x$ in $X$ if, $$d(x_n,x)\equiv ...
0
votes
1answer
29 views

Property of topologically equivalent metrics [closed]

Let $X$ be a compact topological space, and $d_1,d_2$ two metrics that induce the topology on $X$. Is it necessarily true that for every $\epsilon > 0$ there exists a $\delta > 0$ such that: ...
5
votes
2answers
103 views

When a homeomorphism can be upgraded into an isometry?

Let $X$ be a metrizable topological space. Let $f:X \rightarrow X $ be a homeomorphism. When can we find some metric $d$ which induces the original topology on $X$ making $f$ an isometry? Partial ...
1
vote
2answers
30 views

Why is this not a metric on the space of Riemann Integrable functions on [a,b]?

I've just started to teach myself a little on the basics of metric spaces, and came across the following question. Let $d_2$ be the pseudo-metric defined on the space of continuous functions on ...
0
votes
0answers
24 views

A question on the Banach Contraction Mapping Principle

The BCMP states that in a complete metric space $X$, a contraction mapping $T$ on $X$ has a unique fixed point, i.e. if $T$ satisfies $d(Tx, Ty) \le k d(x,y)$ such that $0 \le k < 1$, then $T$ has ...