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
36 views

Which complete weighted graphs are obtained from finite metric spaces?

Let $(X, d)$ be a finite metric space with $X = \{x_1, \dots, x_n\}$. We can associate to this metric space a complete weighted graph with vertices labelled by the points of $X$, and edges weighted by ...
1
vote
0answers
30 views

Order and Metric

Consider the Reals as a totally ordered set via its natural order ( linear continuum ). Such order induces an order topology ( basis is the collection of open-intervals of that order), which ...
0
votes
1answer
26 views

Convergence of sequence of compact sets in Hausdorff metric

Given a sequence of compact sets $K_{i}$ in $\mathbb{R^{n}}$ and a compact set $K$ in $\mathbb{R^{n}}$, which satisfy the following 2 conditions. $\forall$ $x$ $\in$ $K$, $\exists$ $x_{i}$ $\in$ ...
1
vote
1answer
46 views

Why does countable compactness imply compactness on metric spaces?

By "$E$ is countably compact", I mean that every countable open cover of $E$ has a finite subcover. By "$E$ is compact", I mean that every open cover of $E$ has a finite subcover. Let $M$ be a metric ...
1
vote
3answers
143 views

is union of nested compact spaces still compact?

Stel $D$ a metric space. Let $K_1 \subset K_2 \subset K_3 \subset ...$ a serie of compact sets in $D$. I was wondering if $K = \bigcup_{n=1}^\infty K_n$ is compact too. If we take an open cover of $K$ ...
2
votes
1answer
42 views

Total order and its order topology

I noticed that the natural order of the Reals alone, being complete ( satisfying LUB ) , is able to prove that the induced order topology is complete ( every cauchy sequence converges ). We are ...
1
vote
1answer
43 views

Proving set of bounded continuous functions is an open set

appreciate your help with the below: Question: Let C[0,1] be the set of continuous functions from [0,1] to $\mathbb{R}$. Consider the metric space M = (C[0,1],d) where d denotes the sup metric. ...
0
votes
0answers
18 views

$\cup \mathcal C$ is polygonally connected.

Suppose $X$ is a normed linear space and $\mathcal C$ is a chained collection of convex subsets of $X$ then $\cup \mathcal C$ is polygonally connected. A non-empty collection $C$ of sets is said to ...
0
votes
0answers
19 views

All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
0
votes
0answers
13 views

The distance distribution from the mean for an n-dimensional normal(Gaussian) distribution

Let's say we have an n-dimensional normal distribution with identity covariance matrix and 0 mean. When we draw random points in this distribution, how do I get the distribution of the distance from ...
1
vote
0answers
52 views

If $E= A\cup B \cup C$ and $E$ is connected , where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected.

If $E= A\cup B \cup C$ and $E$ is connected in a metric space $(X,d)$, where $A$ and $B$ are disconnected and $C$ is connected, then $A \cup C$ is connected. If we consider that $A \cup C$ is not ...
1
vote
1answer
28 views

Show that the discrete topology on $X$ is induced by the discrete metric

Let $X$ be a set. Show that the discrete topology on $X$ is induced by the metric $d(x, y) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \neq y \\ 0 & \mbox{if } x = y \end{array} ...
1
vote
2answers
55 views

Increasing sequence of open sets in a separable metric space.

Suppose X is a separable metric space and ($U_α$ : α < γ) is an increasing sequence of open sets (i.e. $U_α$ ⊆ $U_β$ for α < β). Show that there is a countable $γ_0$ such that $U_α$ = $U_β$ for ...
1
vote
0answers
20 views

Weak Convergence in Metric Space proof

I have been reading Billingsleys book where I came across this theorem and proof. I am having difficulty understanding the theorem/proof. I feel there is a better, more complete way to prove it. Does ...
-1
votes
1answer
31 views

Distance between a point and an open set in euclidean space

This is one of my Analysis quizzes. Let $S$ is an open subset of $\mathbb{R}^n$. Then the following holds: $$\forall x\in \mathbb{R}^n \backslash S^\circ\; \forall y \in S \backslash \{x\} \quad ...
1
vote
1answer
23 views

Show triangular inequality in metric

Let $(X,\rho)$ be a metric space. I want to show that the function $\sigma(x,y)=\min\{1,\rho(x,y)\}$ is a metric on the set $X$. I It is fairly straightforward to show that $(*)$ ...
0
votes
1answer
22 views

Open set in the sup-metric space

can you please explain why is the following so? Or at least point me in a direction which can help me find the answer? Given a set G of functions g: $\mathbb{R} \rightarrow \mathbb{R}$ such that ...
-1
votes
0answers
17 views

cauchy sequence metric space the same

I have the following question: If $d_1$ and $d_2$ are metrics on the same set $X$ and there are positive numbers $a$ and $b$ such that for all $x$, $y$ in $X$: $$ a \cdot d_1(x, y) \le d_2(x, ...
1
vote
0answers
66 views

$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^0$ , $(\bar A)^0$ , $\overline{A^0}$ are pairwise distinct ? I found an example from a book that such a ...
0
votes
1answer
18 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
1
vote
1answer
30 views

Prove that the given subset satisfying the given hypothesis is compact.

Let C be a subset of a compact metric space (X, d). Assume that, for every continuous function h : X → R, the restriction of h to C attains a maximum on C. Prove that C is compact. My attempt: I ...
14
votes
4answers
1k views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
0
votes
2answers
79 views

Test whether the following sets are connected or not.

Which of the following sets are connected ? (A) $\{(x,y)\in \mathbb R^2:x,y\in \Bbb Q\}\subset \Bbb R^2$. (B) $\{(x,y)\in \Bbb R^2:\text{ at least one of } x,y \text{ is rational ...
0
votes
1answer
33 views

How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
0
votes
1answer
44 views

How to show this is a metric?

$d_1(x_1,y_1)$ and $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about ...
0
votes
1answer
18 views

Prove that if a subset $A$ of a metric space is bounded then the closure of $A$ is bounded and diam(A) is equal to the diam(cl(A)).

Prove that if a subset $A$ of a metric space is bounded then the closure of $A$ is bounded and the diameter of $A$ is equal to the diameter of the closure of $A$. This is the question I am working on ...
1
vote
1answer
39 views

$X$ is Frechet Compact iff $X$ is compact.

I have done the proof that $1)\ X$ is Frechet Compact iff $X$ is sequentially compact. $2) \ X$ is sequentially compact iff $X$ is compact. Thus we can conclude that $X$ is Frechet Compact iff ...
1
vote
1answer
77 views

If $f(x)=\tilde{f}(\|x\|)$ and $f$ is continuous, is $\tilde{f}$ continuous?

I am intrigued by this idea that has come to my mind. Let $f:A\subset\mathbb{R}^n\to\mathbb{R}^n$ be a continous funct, either in a point $x_0\in A$ or in all of its domain $A$, whose values only ...
1
vote
1answer
31 views

Metric and Absolute value function on $\mathbb R$ [closed]

I'm contemplating the notion of the absolute value function on $\mathbb R$ as well as of the usual metric on $\mathbb R$. It seems to me that each one of those can be seen in light of the other. ...
1
vote
3answers
59 views

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic. Please suggest me ways on how should I think about this.Its quite sure that $X$ cant be finite. I tried ...
0
votes
0answers
39 views

Gromov compactness theorem

Reference: this book, page 493. For a compact metric space $X$ define $\text{Cov}(X,\epsilon)= \min \{n \, : \, X \text{ is covered by $n$ closed } \epsilon\text{-balls} \}$ and ...
0
votes
1answer
55 views

Proving that a singleton set is both open and closed inside this metric space

Let $(E,d)$ be a metric space and let $a \in E$. Let $\delta(x,y)=\begin{cases} d(a,x)+d(a,y) & x \neq y \\ 0 & x = y \end{cases}$. It can be proved that $\delta$ is a metric on $E$ (I did ...
1
vote
2answers
54 views

Unions and Intersections of Open Sets are Open

Let $(X,d)$ be a metric space. Prove: the union of any open sets in $X$ is open in $X$ the intersection of a finite number of open sets in $X$ is open in $X$ I could prove the first one but how ...
2
votes
1answer
52 views

To prove Heine-Borel theorem for $\mathbb R^n$ with usual Euclidean topology

To prove that any closed and bounded subset of $\mathbb R^n$ is compact , I proceed as : Since $\mathbb R^n$ is complete so any closed subset of it is complete . Then I show that any bounded subset of ...
5
votes
2answers
43 views

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$.

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$. We assume on the contrary that there does not exist ...
4
votes
1answer
65 views

Maximum number of points you can put on grid $ n\times m$ with no equidistant?

Assume we have a grid of $n\times m$ points. and the distance between two rows or two columns is 1 ( unit ). I have a couple of questions related to this grid:- What is the list of possible length ...
1
vote
1answer
18 views

Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
1
vote
2answers
45 views

Does every subset of a metric space have an open cover?

I'm having some trouble understanding the concept of compact set (I'm studying from Rudin's Principles of Mathematical Analysis). Does every subset of a metric space have an open cover? Why?
1
vote
1answer
74 views

$f(X)$ is uncountable and hence $X$ is uncountable.

My question: let $f : X \to \Bbb R$ be a non-constant continuous function on a connected metric space and assume that $f(X)$ is uncountable; then $X$ is uncountable. We know continuous image of a ...
1
vote
0answers
18 views

small expected contraction embedding into trees?

I learned FRT theorem for probabilistic metric embedding into trees: For any finite metric d, there exists a distribution over non contracting, small expected expansion tree metrics. The theorem can ...
1
vote
1answer
33 views

covering number and compactness

The following picture is what I extracted from the end of page 7 in http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf My confusion is on the blue part: in 1-dimensional ...
2
votes
1answer
28 views

Show that dual space of $R^n$ with norm 3 is equal to the $R^n$ with norm 1.5.

How can one prove that dual space ($R^n$,$||.||_3$)*= ($R^n$,||.||1.5). How to go about using the holder's inequality? Any help will be appreciated! Hint: I know I've to use holder inequality to make ...
2
votes
3answers
66 views

What is wrong with my proof that $f^{-1}(S)$ is open?

Let $X$ and $Y$ be metric spaces, $f: X \to Y$ is continuous, $S \subset Y$, and $S$ open. Prove that $f^{-1}(S)$ is open, where $f^{-1}(S) = \{x \in X : f(x) \in S\}$. If $x \in f^{-1}(S)$, then ...
2
votes
1answer
45 views

A set $A \subset l_1$ is compact

A set $A \subset l_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon >0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| < \epsilon$ for all $n> n_0$ and ...
1
vote
1answer
25 views

$\{U_{\alpha} \}_{\alpha \in I}$ collection of connected sets , for every $U_{\alpha}$ , $\exists U_{\beta}\ne U_{\alpha}$ not mutually disjoint

A probable further strengthening of $\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint ... If $\{U_{\alpha} \}_{\alpha ...
1
vote
1answer
18 views

Closed set and set, closed in $\mathfrak M$

I've read in my textbook that a set $A$ is called closed if it contains its limit points, i.e. $A'\subseteq A$. But then, coming to next chapter, I came across a term of set $B$, closed in metric ...
1
vote
2answers
86 views

Use definitions to show $[0, 1) × [0, 1)$ is neither an open nor closed subset of $\Bbb{R^2}$.

Show, from the definitions of open and closed sets, that when using the standard Euclidean metric, [0, 1) × [0, 1) is neither an open nor closed subset of $\Bbb{R^2}$. From what I understand, a set ...
3
votes
3answers
164 views

Proving Any connected subset of R is an Interval

Common Proof: Suppose $S$ is not an interval of $R$. Then by Interval Defined by Betweenness, $∃x,y∈S$ and $z\in R∖S$ such that $x<z<y$. Consider the sets $A_1=S∩(−∞,z)$ and ...
0
votes
0answers
24 views

Suppose that $f:X \rightarrow X'$ is a one-one correspondence of metric spaces such that $f$ is uniformly continuous and $f^{-1}$ is continuous.

Suppose that $f:X \rightarrow X'$ is a one-one correspondence of metric spaces such that $f$ is uniformly continuous and $f^{-1}$ is continuous. Prove that if $(y_n)$ is a convergent sequence in $X'$ ...
3
votes
0answers
35 views

$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...