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)

2
votes
1answer
34 views

Every metric space contains a discrete, coarsely dense subset

I'm wondering on how to prove the following: Let $(X,d) $ be a metric space. We say that a subset $ A \subset X $ is coarsely dense iff $ \exists_{C > 0} \forall_{x \in X} \exists_{a \in A} d(x,a) ...
4
votes
3answers
65 views

Closest packing of equal balls in $\Bbb{R}^4$

I know how to find the closest packing of equal spheres in $\Bbb{R}^3$. I'd like to know how to find the closest packing of equal balls in $\Bbb{R}^4$ with the standard Euclidian metric. I suspect ...
0
votes
0answers
22 views

Prove that $\frac{d(a,b)}{1 + d(a,b)}$ is a metric? [duplicate]

Given any metric $d(a,b)$ where $a,b\in\mathbb{R}$, prove that $$d_1(a,b):=\frac{d(a,b)}{1+d(a,b)}$$ is also a metric. We have $0 \leq s\leq t \Rightarrow \frac{s}{1+s} \leq \frac{t}{1+t}$ to support ...
0
votes
2answers
22 views

Open set in a metric space is union of closed balls?

We know that every open set $A$ is in a metric space $(X,d)$ is the countable union of closed sets, and every open set $A$ is in a Euclid space $R^n$ is the countable union of closed balls. My ...
1
vote
1answer
26 views

Inequality involving distance between two points

Let $\Omega$ be a set in $\mathbb{R}^n$ Fix $x\in\Omega$, and $p\notin \Omega$ Then, is it always true that $$\|x-p'\| \leq \|x-p\|$$ , where $\|\cdot\|$ is the Euclidean norm, and $p'$ is the ...
0
votes
2answers
45 views

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$

Show that $\mathbb{R}$ and $\mathbb{R}^n$ are not homeomorphic if $n\geq 2$. I want to use a connection-type argument. I thought of giving the following proof; Suppose that there exist such a ...
1
vote
1answer
29 views

The space of absolutely convergent series is complete

For clarity: $$ l^{1}(\mathbb{N}) = \left\{ (x_{n})_{n} \ \middle|\ \sum_{n}|x_{n}| \in\mathbb{R} \right\} $$ $$ d_{1}:\ l^{1}(\mathbb{N}) \times l^{1}(\mathbb{N}) \rightarrow \mathbb{R}^{+}:\ ...
0
votes
1answer
18 views

How to convert table into a distance function?

Been stumped on this past paper question for a while, it's in the context of clustering and the next part is using single linkage bottom-up hierarchical clustering to form a dendrogram using your ...
0
votes
1answer
36 views

Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where ...
0
votes
1answer
13 views

A convergent sequence of non-expansions converges uniformly on a totally bounded domain

Here's a theorem that I tried to prove: Let $V,d_V$ and $W,d_W$ be metric spaces and $(f_n)_n$ a sequence of non-expansions that converges to a function $f:A \subseteq V \rightarrow W$: $$ f_n:\ A ...
1
vote
1answer
35 views

Can I pull a limit through a metric?

Let $X,d_X$ and $Y,d_Y$ be metric spaces and $(f_n)_n$ a sequence of (continuous) functions. Does this hold and, more importantly, why? $$ d_Y\left(\lim_{n\rightarrow +\infty}f_n(x), ...
0
votes
1answer
24 views

Book recomendation for function sequences.

I wanted to study about sequences of functions defined in metric spaces. What book/books do you recommend? Thanks!
3
votes
2answers
56 views

In a metric space $(X, d)$, if closed sets $A$, $B$ contain sequences $a_n,b_n$ such that $d(a_n,b_n)\to 0$, must $A\cap B\neq \emptyset$?

I wrote a test yesterday in which one of the questions asked us to prove that if $A$ and $B$ are disjoint closed subsets of a metric space $(X, d)$, then there exist disjoint open subsets $U$ and $V$ ...
-1
votes
2answers
25 views

If $C \subset X$ and $\mathcal{U} \subset X$ is open. Is $C \cap \mathcal{U}$ open in $C$?

In an exercise regarding connection, I came to the following problem, I am given $C \subset X$ and $\mathcal{U} \subset X$ is open (where $(X,d)$ is a metric space). And I could use that is $C \cap ...
0
votes
1answer
15 views

Conceptual question regarding conection in metric spaces.

I have to give an example of two sets $A,B \subset \mathbb{R}$ such that both are connected, but $A\cup B$ is not. So I thought of a trivial example $(0,1) \subset \mathbb{R}$, $(2,3) \subset ...
1
vote
1answer
14 views

Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$

Let $M$ be a metric space, let $K \subset V \subset M$, $K$ compact, $V$ open. Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$ I came up with a proof, but there is ...
0
votes
1answer
14 views

Distance attained by a function

Let $A$ be a subset of $\mathbb R^n$ and let $x\in \mathbb R^n$. Then $\exists y_0\in A$ such that $d(x,y_0)=d(x,A)$ if $A$ is a non-empty subset of $\mathbb R^n$. $A$ is a non-empty closed subset ...
3
votes
2answers
34 views

Is this function a metric?

Let $X$,$d$ be a metric space. Define $d'$ as the minimum of $1$ and $d$: $$ d':\ X^2 \rightarrow \mathbb{R}:\ d'(x,y) = \min\{1,d(x,y)\} $$ The question is whether $d'$ is a metric. I've managed to ...
3
votes
3answers
111 views

$[0,1)$ as a subspace of the Euclidean metric space?

Consider the Euclidian metric space $(\Bbb R,d)$ where $d(x,y)=|x-y|$ is the usual metcic on $\Bbb R$. The set $[0,1)$ is not closed in $(\Bbb R,d)$ but considered as a subspace it is closed by ...
3
votes
1answer
38 views

Two problems related to continuity of a metric from Munkres' topology book

Let $X$ be a metric space with metric $d$. Show that $d:X\times X\to \mathbb{R}$ is continuous. Let $X^\prime$ denote a space with the same underlying set as $X$. Show that if $d:X^\prime\times ...
3
votes
1answer
23 views

Compactness of a group with a bounded left-invariant metric

Let $G$ be a group equipped with a left-invariant metric $d$: that is, $(G,d)$ is a metric space and $d(xy,xz) = d(y,z)$ for all $x,y,z \in G$. Suppose further that $(G,d)$ is connected, locally ...
-2
votes
2answers
63 views

Short proof that $\rho^\prime(x,y) = \min\{1,\rho(x,y)\}$ is a metric [duplicate]

Let $(X,\rho)$ be a metric space. Define $\rho^\prime: X \times X \to \mathbf{R}$ by $\rho^\prime (x,y) = \min\{1,\rho(x,y)\}$ for all $x, y \in X$. Does anyone know of a short proof that ...
0
votes
2answers
38 views

Compactness of $A\subset \mathbb R$ w.r.t. two different topologies

Let $d$ be the Euclidean metric and $d'$ be any other metric on $\mathbb R$. Let $A\subset \mathbb R$ be a closed and bounded subset with respect to $d'$. Then which is TRUE ? (A) $A$ is ...
1
vote
1answer
37 views

A fixed-point theorem by Zamfirescu

I am having a trouble with understanding the proof of a fixed-point theorem by Zamfirescu. Could somebody please explain how the inequality in the inner, pink rectangle is obtained from the previous ...
0
votes
1answer
57 views

Show that A is an open subset of M

If $m\in\mathbb{N}$, $M=\{0,1\}^{\mathbb{N}}$ and $A \subseteq M$ is an open set of sequence where the number 1 appears at least $m$ times. Show that $A$ is an open subset of $M$. I wanted to show ...
1
vote
1answer
19 views

Is the closure of a meager set meager?

How to show that the closure of a meager set is meager? I tried like this: Suppose that it is not meager then $cl(A)$, where $A$ is a meager set in a metric space $(X,d)$ contains an interior point ...
0
votes
0answers
25 views

Closed subsets of empty interior are of 1st category

In a metric space is it true that closed sets with empty interior are of 1st category? I.e., that it can be represented as a at most countable union of meager sets? Thanks
0
votes
1answer
35 views

Is point-to-set distance function $C^\infty$ for $\mathbb{R}^n$

Let $x\in \mathbb{R}^n$ and $Q\subset \mathbb{R}^n$. Then we define the point-to-set distance function as: $$ d_Q(x) = \inf_{y \in Q} \| x-y\| $$ It's continuous for every normal space (not only ...
0
votes
0answers
23 views

Holder's inequality/Cauchy-Schwartz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) ...
3
votes
2answers
71 views

Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
2
votes
2answers
32 views

Prove that $\delta$ is a metric in $\mathcal{K}(X)$

Let $(X,d)$ be a complete metric space. We define $\mathcal{K}(X)=\{K \subset X : K \text{ is compact and non empty}\}$ Define $d'(A,B)=sup_{a \in A}\{d(a,B)\}$ Show that $\delta$ ...
1
vote
1answer
13 views

how to prove the sequence based definition of a closure in metric spaces

Let $A$ be a subset of a metric space $\Omega$. By definition, the closure of $A$ is the smallest closed set that contains $A$. How to prove that alternativelly, the closure is given by (1): $\bar A ...
1
vote
1answer
38 views

Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
0
votes
0answers
44 views

Does it make sense to apply the Manhattan metric to an arbitrary graph?

I overheard someone talking about using the Manhattan metric against nodes on an arbitrary graph (or even a tree). At the time, I didn't think much of it, but having dwelled upon it, does it even make ...
2
votes
2answers
57 views

Confused about proof that diameter of a closure of a set is the same as the diameter of the set.

Definition Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$, with $p \in E$ and $q \in E$. The supremum of $S$ is called the diameter ...
4
votes
0answers
43 views

Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...
2
votes
2answers
23 views

Show that the closed ball $B[x,1]$ in $c_0$ is not compact.

Consider $c_0 = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}:a_n \to 0\}$. Show that the closed ball $B[x,1]=\{y \in c_0 : d_{\infty}(x,y)\leq 1\}$ is not compact in $c_0$. Were ...
2
votes
1answer
22 views

Question about disconnected metric spaces

The definition of disconnectedness that I've been taught is that a metric space $(X,d)$ is disconnected if there exists two non-empty disjoint open sets $A$ and $B$ such that $X=A\cup B$. My ...
0
votes
0answers
24 views

Metric invariant under translations is projective

Show that a metric d on the plane that is invariant under translations is automatically projective. Note that we only consider length spaces, which are metric spaces where the distance between two ...
3
votes
1answer
35 views

Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
2
votes
2answers
41 views

Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$

Let $(X,d)$ be a metric space. Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$. If $F\subseteq X$ is closed then $K\subseteq X$ compact implies $K$ ...
1
vote
1answer
16 views

p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
4
votes
1answer
94 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
4
votes
2answers
37 views

On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...
1
vote
1answer
28 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
0
votes
1answer
27 views

If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
2
votes
2answers
135 views

$\epsilon$-dense subsets on $\mathbb R/\mathbb Z$.

Let $\langle M, d\rangle$ be a metric space. We say that $A \subset M$ is $\epsilon$-dense if every open ball of radius $\epsilon$ contains a point of $A$. Now let $T=\mathbb R/\mathbb Z$, the ...
2
votes
1answer
28 views

Sufficient conditions for embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.

This is a followup to a question I asked in this thread. I'm posting separately so points can be awarded. Hopefully someone can help me with a reference for this problem, or the construction. I ...
8
votes
2answers
187 views

On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive-definite $n\times n$ real matrices. I am looking for hints concerning the visualization of such spaces for $n=1,2,\ldots$. I know that ...
1
vote
1answer
28 views

Two exercises about hypermetric spaces

Take $S$ to be the collection of all subsets of $\{1,\dots,n\}$. If $x, y$ are in $S$, define $d(x,y)$ as the number of elements of the symmetric difference $x\triangle y$. Exercise 2.1. Show ...