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
0answers
12 views

If $(X,d_1)$ and $(X,d_2)$ two connected metric spaces if only if $X\times Y$ is connected metric space

$(X,d_1)$ and $(X,d_2)$ are two connected metric spaces if and only if $X\times Y$ is a connected metric space with metric $$ D((x_1,y_2), (x_2,y_2)) = \max(d_1(x_1,x_2),d_2(y_1,y_2)).$$ I know that ...
0
votes
0answers
9 views

Preservation of completeness through a continous onto mapping

Let $(X_{1},d_{1})$ and $(X_{2},d_{2})$ be metric spaces and $f: X_{1} \to X_{2}$ be a continuous onto map such that $$ d_{1}(x,y) \leq d_{2}(f(x),f(y)) \hspace{2mm} \forall\phantom{i}x,y \in ...
0
votes
2answers
19 views

Interior of a set in a metric space

if $E$ is a metric space nd $B\neq E$ how to prove that: $$\overset{\circ}{B}=\bigcup_{n=1}^{\infty} (\{x\in E, d(x, E\setminus B)\geq \frac1n\})$$ i don't know how to start
0
votes
1answer
10 views

Proving that a set infers a norm given certain conditions

Let $\bar{B}$ be a set in a vector space $E$. Then $\bar{B}$ is the closed unitary ball for a norm iff $\mathbb{N} . \bar{B} = E$ $\lambda x + (1-\lambda)y \in \bar{B}$ for all $x \, ,y \, \in ...
1
vote
1answer
9 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
0
votes
1answer
17 views

For $U\subseteq Y\subseteq X$, prove that $U$ is open in $Y$ iff there is a $V\subseteq X$ such that $U=Y\cap V$

Let $(X,d)$ be a metric space, with $Y$ a subset of $X$. How do I prove that a subset $U\subseteq Y$ is open in the metric space $(Y,d|_{Y\times Y})$ iff there exists an open subset $V$ of $X$ ...
0
votes
1answer
37 views

If $\sum_{n=1}^{\infty}x_n^2<\infty$ and $\sum_{m=1}^{\infty}x_n^2<\infty$, is $\sum_{k=1}^{\infty}(x_n)_k^2(x_m)_k^2<\infty$? [duplicate]

Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$ equipped with the norm $$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$ Prove that $l^2$ is complete with ...
0
votes
2answers
23 views

Prove that all three metrics induces the same topology on $X_1\times X_2$

Prove that if $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces on $X_1\times X_2$ and metric $d:(X_1\times X_2)\times (X_1\times X_2)\rightarrow R$ is defined in following way: ...
1
vote
2answers
46 views

How to resolve the apparent paradox resulting from two different proofs?

Definition of Open Ball Let $(X, d)$ be a metric space and let $r\in\mathbb{R}^+$. Then the set, $B_d(x, r) := \{y \in X : d(x, y) < r\}$ will be said to be the open ball of radius $r$ ...
6
votes
4answers
136 views

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in ...
1
vote
1answer
31 views

Function between two metric spaces?

I need to come up with: two metric spaces ( X , d ) and ( Y , p ) A continuous function f: X → Y A Cauchy sequence {xn} in X that isn't mapped to a Cauchy sequence in Y My idea was to make ...
3
votes
2answers
35 views

Map from circle to real line

I am asked to show that, for any continuous $\phi:\;S^1\to\mathbb{R}$ where $S^1=\{ \|\mathbf{x}\|=1,\;\mathbf{x}\in\mathbb{R}^2\}$, there exists $\mathbf{z}\neq 0$ such that: ...
0
votes
2answers
25 views

What values of $b$ such that $f_n(x)=b\cos\left(\frac{x}{n}\right)$ converges uniformly?

For what values of $b$ does the sequence of functions: for each $n\in\mathbb{N}$, let $$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$ converge uniformly in the space $C[0,1]$ equipped ...
1
vote
0answers
32 views

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
0
votes
2answers
55 views

A sequence of functions converges in $C[0,1]$ iff it is Cauchy? Is it pointwise or uniform convergence?

In my notes, there is this theorem: A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy. I understand that this theorem applies to all complete metric spaces, not just to $R^n$. ...
2
votes
2answers
36 views

Any necessary and sufficient condition(s) for closure of an open ball to be the corresponding closed ball?

Let $(X,d)$ be a metric space, $a \in X$, and $\delta$ be a positive real number. Then the open ball $B(a;\delta)$ is defined as $$B(a;\delta) \colon= \left\{ \ x \in X \ \colon \ d(x,a) < \delta ...
0
votes
0answers
22 views

subset of a complete space has a relatively compact $\varepsilon$-net.

Trying to prove that, a subset $A \subset X$ of a complete space $X$ is relatively compact iff $\forall \epsilon > 0$ $A$ has a relatively compact $\epsilon$- net. I have proved the following ...
1
vote
2answers
37 views

Definition of topological space

The definition of a topological space is a set with a collection of subsets (the topology) satisfying various conditions. A metric topology is given as the set of open subsets with respect to the ...
0
votes
0answers
26 views

Can we define the derivative of a function in arbitrary metric space in the following way?

Let us first define some terms. Definition of Pre-pseudometric Let $X\ne\emptyset$ and a function $\varphi:X\times X\to\mathbb{R}$ will be called a pre-pseudometric on $X$ if, ...
0
votes
1answer
22 views

Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
0
votes
1answer
21 views

How can I show that the sequence $a_n := p^n$ is a convergent sequence in this metric and find its limit?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
1
vote
0answers
8 views

Equicontinuous homotopies of families of uniformly equicontinuous functions

Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of ...
1
vote
1answer
14 views

Is there a term for this line like subset within a metric space

While thinking about geodesic lines I started exploring subsets of a metric space that have the following property. $ \forall a,b,c \in L, d(a,c)> d(a,b) \land d(a,c) > d(b,c) \implies d(a,c) = ...
0
votes
2answers
19 views

If $Z(f)$ is the zero set, prove that $Z(f)$ is closed

Introduction: Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98. Exercise: Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of ...
3
votes
1answer
72 views

Is the empty set an open ball in a metric space?

Problem Let $(X,d)$ be a metric space where $X$ is a non-empty set. Is the empty set an open ball in $X$? I think that it is true because if $X=\mathbb{R}$ with the usual metric then for all ...
2
votes
2answers
54 views

proof that an arbitrary homeomorphism $h: B_{1}[0] \rightarrow B_{1}[0]$ maps $S^n$ to $S^n$

Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove. Is there any simple proof (or counter-example) for the proposition: Consider the closed ball of ...
1
vote
1answer
36 views

EDITTED: Find all values of $a$ and $b$ so that $ax^n+b\cos\left(\frac{x}{n}\right)$ is Cauchy.

For each $n\in\mathbb{N}$ let $$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$ Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of ...
3
votes
1answer
14 views

Any bi-invariant distance on a group is inverse-invariant?

$\newcommand{\inv}{\text{inv}}$ Let $G$ be a Lie group. Assume $d$ is a metric on $G$ (in the sense of metric spaces) which is bi-invariant. Is it true that the inverse automorphism must be an ...
0
votes
0answers
51 views

Is every compact metric space finite dimensional?

It certainly is true if the metric space is normed by Riesz lemma.
0
votes
1answer
27 views

Equivalent Metrics on $\mathbb{R^n}$

I am working on a problem and want to verify that my logic and reasoning is correct. This is my first time working with metric spaces. Show that the following define equivalent metrics on ...
2
votes
0answers
32 views

Does a mapping from one metric space to another metric space preserve star-likeness of regions?

Let $X$ be a vector space, let $M_1 = \left(X, d_1\right)$ be a metric space and let $M_2 = \left(X, d_2\right)$ be another one. $f : M_1 \to M_2$ is continuous and the origin is a fixed point. $f$ ...
0
votes
1answer
14 views

Is a metric space a requirements for the application of the algebra of events from probability?

When I refer to a metric space, I mean a space that has some genuine notion of distance. In some applied context, this distance would be computed with respect to a coordinate system. I just wanted to ...
0
votes
2answers
32 views

What is the connection between $l_p$ norms and “$l_p$ metrics”?

In some textbooks metric spaces you sometimes encounter these "$l_p$ metrics", $d_1, d_2, d_\infty$ (I don't think $l_p$ metric is very standard usage) For example, $d_1(x,y) := \sum\limits_i^m ...
0
votes
0answers
18 views

How to perform statistical test for two sets of points?

(I have asked this question originally on Cross Validated; however, no good answer and someone suggested me to ask the question here). Thanks a lot in advance if anyone can help. We know that we can ...
0
votes
0answers
13 views

Grassmannian Non-Convex

The Grassmannian manifold $Gr(r,V)$ defines the set of $r$-dimensional linear subspaces of the vector space $V$. My question is, in general, what is the simplest way to see that $Gr(r,V)$ is a ...
1
vote
2answers
27 views

Showing interior of a set is empty

Consider the metric space $(C[0,1],d_{\infty})$. For $x_{0} \in [0,1]$ and $M > 0$ define the set $A \subset C[0,1]$ by $$ A = \{f \in C[0,1] \phantom{.}|\phantom{.} |f(x) - f(x_{0})| \leqslant ...
1
vote
1answer
23 views

Trouble understand a step in the proof that $l^p$ is complete

I'm reading through a proof, attached here. I didn't include the whole proof. The last step is the one I'm confused about. Shouldn't there be more of a justication for taking $\lim_{n \to \infty}$ ...
1
vote
1answer
20 views

Distance function is continuous in topology induced by the metric

The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ ...
3
votes
1answer
62 views

Is this set open in the product topology?

Let $X$ and $Y$ be topological spaces and equipp $X\times Y$ with the product topology. Assume $U\subset X$ is open and for every $p\in U$ we have an open subset $V_p\subset Y$ of $Y$. Is the set ...
-1
votes
0answers
37 views

Topological Equivalence of Metric Spaces [closed]

Suppose we have two different metric spaces $(X,\phi)$ and $(Y,\psi)$. I need to show that the metrics $\phi$ and $\psi$ are equivalent metrics. Using a sterographic projection, I've shown that if we ...
2
votes
2answers
39 views

Product spaces and open sets

I have a proposition I have been pondering that I need help with. Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. Recall that the product space $(X\times Y, d_{1})$ is also a metric space with the ...
0
votes
0answers
20 views

Creating a configuration of points where each point is away from all other points by a pre-defined distance

Let's assume that the points $\in \mathbb{R}^2$ and there are only C=5 points (in practice, I may have $\mathbb{R}^{800}$ and 1000 points). The first out of the five points is fixed. We also have been ...
0
votes
2answers
76 views

Topology - Open set [closed]

How do I show this? Let $(X,d)$ be a metric space and $x\in X$ an element in $X$. Show that \begin{align*} \{y\in X|d(y,x)>r\} \end{align*} is open for any $r\in \mathbb{R}$. The definition of a ...
3
votes
1answer
50 views

Is it always possible to find a “pre-metric” from a metric?

Problem Let $X$ be a non-empty set. Let $f:X\times X\to \mathbb{R}$ satisfying the following properties, $f(x,y)=0\iff x=y$ for all $x,y\in X$. $f(x,y)=-f(y,x)$ for all $x,y\in X$. ...
1
vote
0answers
28 views

Are the set of probability functions with compact support in a fixed closed ball complete under the Wasserstein norm?

Let $B_R$ be a closed ball of radius $R$ in the space $\mathbb{R}^d$. As the title suggests I have this feeling that the set of functions $$S:= \left\lbrace f:\mathbb{R}^d \to \mathbb{R} ...
2
votes
2answers
46 views

Theorem 2.41 in Baby Rudin: Is this proof good enough? Can we generalise it?

Here is Theorem 2.41 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If a set $E$ in $\mathbb{R}^k$ has one of the following three properties, then it has the other ...
0
votes
0answers
18 views

Are sequences themselves metric spaces with the inherited metric?

I have been asked to show whether a sequence $(p_n)$ in $\mathbb{R}$ is a metric space with the inherited metric $d(x,y) = |x-y|$ It seemed to me at first to be a slightly odd question because we ...
2
votes
1answer
53 views

If two nested open sets have the same nonempty boundary, are they the same set?

Let $(X,d)$ be a metric space. Let $B_\epsilon(x)$ be the open ball of radius $\epsilon$ centered at $x$. For $x\in X$ and $\epsilon>0$, suppose that $V$ is an open set in $X$ with $V\subseteq ...
1
vote
3answers
38 views

Subset of $(l^{2},d_{2})$ is open

Show that $A = \{\phantom{i}\{x_{n}\} \in l^{2} \hspace{2mm}:\hspace{2mm} |x_{n}| < 1, \forall \phantom{i}n \in \mathbb{N}\phantom{i} \}$ is open in $(l^{2},d_{2})$. The $d_{2}$ metric is: $$ ...
2
votes
1answer
45 views

Theorem 2.27 (a) in Baby Rudin: Is his proof complete enough?

Here's Theorem 2.27 (a) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $X$ is a metric space and $E \subset X$, then $\overline{E}$ is closed. Now here's ...