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)

0
votes
1answer
30 views

Problem in showing that a sequence is a Cauchy sequence on a space with the integral metric.

I'm having difficulty following what is going on and understanding parts in the following example. It is quite similar to an example I posted before (Changing of the limits of integration with the ...
1
vote
2answers
36 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
1
vote
4answers
77 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
1
vote
1answer
31 views

The convergence of different metrics on the same space

The following example is from my notes, and I would like clarification on some wider points connected to it, namely about extensions from what we understand metrics and metric spaces to be. It follows ...
2
votes
1answer
61 views

Prove that a finite union of closed sets is also closed (using limit points)

Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed. Proving that statement is equivalent to proving: If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then ...
1
vote
0answers
28 views

When does equality occur in the triangle inequality in metric space? [on hold]

When I think of $\mathbb{R}^n$ , $n\leq 3$ ; it is very easy given the usual metric. But what if the metric is not usual? I cannot think of a metric which is not usual; and still gives an equality ...
3
votes
0answers
29 views

Defining a metric in the tangent spaces $ T_xM $

I'm working with a metric $D$ over the manifold of grassman $G_n(\mathbb{R}^{d})$ and have difficulties to extend $D$. Let me explain: If $M$ is a submanifold of dimension $n$ in $\mathbb{R}^{d}$ ...
2
votes
4answers
55 views

Prove $\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2$

If $X,Y$ are vectors in $\mathbb{R}^n$ and $a>0$ show that: $$\left|\sum_{i=1}^n x_i y_i \right| \le \dfrac{1}{a} \sum_{i=1}^n {x_i}^2 + \dfrac{a}{4}\sum_{i=1}^n {y_i}^2 (*)$$ I started with ...
3
votes
1answer
32 views

Complement of the union of countably many , mutually disjoint , non-empty open balls in $\mathbb R^n , (n >1) $ is path connected?

Let $n \ge 2$ and $\{B_m\}_{m=1}^\infty$ be countably infinitely many , mutually disjoint , non-empty open balls in $\mathbb R^n$ , then is $\mathbb R^n \setminus \cup_{m=1}^\infty B_m$ ...
1
vote
1answer
27 views

Changing of the limits of integration with the integral metric.

Consider the following sequence of functions, $$f_n(x) = \begin{cases} nx & \text{for $0\le x \le \frac1n$} \\ 1 & \text{for $x\ge \frac1n$} \end{cases}$$ And call to mind the integral ...
1
vote
2answers
32 views

Predicate logic inference in a simple proof of uniform continuity.

For a function $f$ from a metric space $X$ into a metric space $Y$, uniform continuity can defined in this way: $\forall ε>0:\existsδ > 0:\forall p,q\in X:d_{X}(p,q)<δ \rightarrow ...
3
votes
0answers
19 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
0
votes
1answer
37 views

Showing $d(x,y)=0$ iff $x_{n}=y_{n}$

Consider the space $\mathbb{R}^{\infty}$ of all sequences $x=\left \{ x_{1},x_{2},... \right \}$ of real numbers. Define the function $d:\mathbb{R}^{\infty}\times \mathbb{R}^{\infty}\rightarrow ...
-3
votes
2answers
26 views

$A$ is a convex subset with non-empty interior and $D$ is dense in $\mathbb R^n$ ; then $\mathbb R^n$ , $U\cap D \cap A \ne \phi$? [on hold]

Let $A$ be a convex subset , with non-empty interior , of $\mathbb R^n$ and $D$ be a dense subset of $\mathbb R^n$ ; then is it true that for every open subset $U$ of $\mathbb R^n$ , $U\cap D \cap A$ ...
0
votes
3answers
12 views

$A$ be a convex subset and $D$ be dense in a real NLS ; then for every non-empty open subset $U$ of the NLS , $U\cap D\cap V \ne \phi$?

Let $A$ be a convex subset of a real normed linear space $V$ and $D$ be a dense subset of $V$ ; then is it true that for every non-empty open subset $U$ of $V$ , $U\cap D\cap A $ is non-empty ?
0
votes
2answers
174 views

Isometry of a metric space with proper subset

In Irving Kaplansky's "Set Theory and Metric Spaces", exercise 17 on page 71 asks for an example of a metric space which is isometric to a proper subset of itself. Any infinite discrete space and any ...
1
vote
2answers
394 views

a bijection between the set of all open sets in M and the set of all closed subsets of M, where M is a metric space

Let T be the collection of open subsets of a metric space M, and K the collection of closed subsets. Show that there is bijection from T onto K. Here, I was thinking that if $f: T\rightarrow K $ is ...
1
vote
1answer
48 views

Confused about the open/closed set in metric space

Let $(M,d)$ be a metric space. I understand well that $\emptyset$ and $\mathbb{R}$ are both open and closed sets. I read some notes that say, that $\emptyset$ and $M$ are both open and closed. So, ...
12
votes
0answers
157 views

Global structure of the Gromov-Hausdorff space

EDIT: now crossposted at mathoverflow (http://mathoverflow.net/questions/212364/on-the-global-structure-of-the-gromov-hausdorff-metric-space) This is a purely idle question, which emerged during a ...
4
votes
1answer
62 views

Why is the Gromov-Hausdorff distance a metric?

The Gromov-Hausdorff distance is: $$ d_{GH}(A,B) = \inf_{f,g}d_H(A',B') $$where $f$ and $g$ are isometric embeddings of $A,B$ into some metric space, and their images are $A', B'$. The inf is taken ...
3
votes
1answer
43 views

Distance between a point and an empty set: meaning and value?

On page 253 in General Topology by R Engelking: The distance $\rho(x, A)$ from a point $x$ to a set $A$ in a metric space $(X,\rho)$ is defined by letting $\rho(x, A) = \text {inf}\ {\{\rho(x, a) ...
2
votes
3answers
90 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
4
votes
4answers
2k views

Prove that a subset of a separable set is itself separable

The problem statement, all variables and given/known data: Show that if $X$ is a subset of $M$ and $(M,d)$ is separable, then $(X,d)$ is separable. [This may be a little bit trickier than it looks - ...
0
votes
0answers
26 views

Statement about the discrete (metric) space, and both an open and closed ball.

I have the following statement from my notes: "Let $(X,d)$ be the discrete space i.e. any non-empty set with the discrete metric ($d_d(x,y)=1$ for all $x\neq y$). Then, amazingly, $B_1(x)=\{x\}$, a ...
0
votes
1answer
38 views

Dense $G_{\delta}$ set implies comeagre set

Suppose that $X$ is a metric space. Show that if $D$ is a dense $G_{\delta}$ set, then $D$ is comeagre, that is, countable intersection of dense sets. My attempt: Let $D=\bigcap_{n \in ...
1
vote
0answers
13 views

Find Connected Components

Let $X = \mathbb{R^2}$ and let $F_k$ be the closed line segment joining $(-1,2^{-k})$ and $(1,2^{-k})$ Additionally let $a = (-1,0) \ ,\ b=(1,0)$ Consider $A = \{a,b\} \cup \ ...
1
vote
1answer
315 views

Show for $f:A \to Y$ uniformly continuous exists a unique extension to $\overline{A}$, which is uniformly continuous

Working on the following problem from Munkres: Let $(X, d_{X})$ and $(Y, d_Y)$ be metric spaces; let $Y$ be complete. Let $A \subset X$. Show that if $f:A \to Y$ is uniformly continuous, then ...
1
vote
2answers
558 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 ...
3
votes
3answers
55 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
0
votes
2answers
61 views

If $d_1(x,y)$ and $d_2(x,y)$ are metrics, prove that $d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$ is a metric.

$$d'(x,y)= \sqrt{d_1^2(x,y)+d_2^2(x,y)}$$ The first three properties are trivially proven. The triangle inequality, not so much. I tried using the triangle inequalities that apply to $d_1$ and $d_2$, ...
0
votes
1answer
30 views

Locally lipschitz mapping implies lipschitz over compacts

Let $f: M_{1} \times M_{2} \to M_{3}$ be locally lipschitz in the first variable. Prove that, given a compact $N \subset M_{1} \times M_{2}$, the mapping $f$ is lipschitz in the first variable over ...
4
votes
1answer
765 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 ...
1
vote
1answer
30 views

Quadratic form as generalized distance?

In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin: $$x^{T}x = ||{x}||^2$$ which represents the square of the ...
-1
votes
3answers
43 views

Constructing true metrics in infinite dimensional vector spaces?

Is there an example of a true metric defined on a function space? I'd imagine it is some type of integral involving two functions, and it will return a value that obeys the metric axioms, but I have ...
1
vote
1answer
63 views

What is wrong with my brute-force approach to proving that $\mathbb R$ as a metric space obeys the triangle inequality?

In a self-study of metric spaces, I'm looking at the very basic exercise of proving that $(\mathbb R, |y-x|)$ is a metric space. The sticking point was the triangle inequality. I did manage to ...
1
vote
1answer
14 views

Lipschitz maps on locally compact groups

Suppose $G$ is a locally compact second countable group. This means that there exists a proper (closed bounded sets are compact) left invariant ($d(gx,gy) = d(x,y) \ \forall g,x,y \in G$) metric on ...
0
votes
1answer
20 views

Inherited properties of boundary points [closed]

My question is related to subspaces of metric spaces, and the inherited properties of points in those subspaces. Given a subspace $(D,d) \subset (X,d)$, where $d$ is a well defined metric. Say that ...
0
votes
1answer
34 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
0
votes
1answer
41 views

How to show $G$ is a perfect set that contains no rational points?

For $E:=[0,1]$, since $\Bbb Q\cap E$ is enumerable, let it be $\{q_1,q_2,\cdots\}$. If I remove the elements of $V_1:=(q_1-\frac1{10},q_1+\frac1{10})$ from $E$, I obtain a closed (and compact) set ...
1
vote
1answer
28 views

Prove the Supremum is attained.

Let $F$ denote denote the set of real valued functions on $[0,1]$ such that, 1) $ \; |f(x)| \leq 1 \; \forall x \; \in [0,1]$ 2) $ \; |f(x)-f(x')| \leq |x-x'| \; \: \forall x,x' \: \in [0,1] $ ...
0
votes
0answers
10 views

Finiteness of Assouad-Nagata dimension in neighbourhood

Given a metric space $Y \subset X$ with finite Assouad-Nagata dimension. When is it possible to find a $\epsilon$-neighbourhood (meaning $N_\epsilon = \{x \in X : d(x,Y) \leq \epsilon\})$ which also ...
0
votes
1answer
14 views

Maximal subset with finite Assouad-Nagata dimension

Given some space $X$ with non-finite Assouad-Nagata dimension. Is it possible for a subset $Y \subset X$ with finite Assouad-Nagata dimension to exist such that $Y$ is maximal in the sense that if any ...
1
vote
0answers
25 views

Using CAT(0) inequality

Let $X$ be a CAT(0) space with metric $d$. Let $p,x,y$ three points on $X$, and let $u,v$ be points on geodesic $[p,x]$ and geodesic $[p,y]$ such that $d(p,u)\geq a,d(p,v)\geq a$,where a is some ...
1
vote
2answers
18 views

Proving that a subset endowed with the discrete metric is both open and closed - choice of radius of the ball around a point

My question is related to proving that any subset $D \subset X$, where $(X,d)$ is a metric space with $d$ being the discrete metric, is both open and closed. I've read some suggestions to a ...
1
vote
1answer
298 views

Continuity of distance function and its generalization

The starting is an easy undergraduate problem. The distance function $d: X \times X \rightarrow \mathbb{R}$ in a metric space $(X,d)$ is continuous. Please check if my proof is correct. If it is wrong ...
0
votes
3answers
78 views

Showing this metric space is complete

Let $X=(0,1]$ and $d(x,y)=\left|\frac{1}{x}-\frac{1}{y}\right|$. I've proven $(X,d)$ is a metric space but I don't know how to show its completeness. How can I do that?
2
votes
1answer
56 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
2
votes
2answers
15 views

Suppose $X$ is a metric space. Let $C$ denote the collection of all dense subsets of $X$. Show that $\bigcap C = iso(X)$

Suppose $X$ is a metric space. Let $C$ denote the collection of all dense subsets of $X$. Show that $\bigcap C = \mathrm{iso}(X)$, where $\mathrm{iso}(X)$ refers to the set of all isolated points of ...
2
votes
2answers
38 views

How do I rate smoothness of discretely sampled data? (Picture!!!)

In the sense that the following curves pictured in order will be rated 98%, 80%, 40%, 5% smooth approximating by eye. My ideas: (1) If the curves all follow some general shape like a polynomial ...
-1
votes
1answer
34 views

Proving that is $A:X \implies Y$ is a linear operator from metric space X to Y is continuous iff it is bounded bounded

The $\implies$ part interests me. The proof given goes like this: Let $A$ be continuous in 0 (because the 0 vector is in every vector space) $B_y(0,r)=\{y \in Y | \| y\|<r \} \implies \exists ...