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)

18
votes
3answers
2k views

Continuity of the function $x\mapsto d(x,A)$ on a metric space

Let $(X,d)$ be a metric space. How to prove that for any closed $A$ a function $d(x,A)$ is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof: $$ |d(x,a) - ...
67
votes
7answers
8k views

$\pi$ in arbitrary metric spaces

Whoever finds a norm for which $\pi=42$ is crowned nerd of the day! Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way? For ...
1
vote
1answer
4k views

A isometric map in metric space is surjective? [duplicate]

Possible Duplicate: Isometries of $\mathbb{R}^n$ Let $X$ be a compact metric space and $f$ be an isometric map from $X$ to $X$. Prove $f$ is a surjective map.
12
votes
5answers
704 views

Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
7
votes
2answers
3k views

If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved

Let $A, B$ be subsets of a metric space $X$. If $A$ is compact and $B$ is closed, show that the distance between $A$ and $B$ is achieved. Attempt at a proof: Let $A$ be compact and $B$ be ...
5
votes
1answer
2k views

Every subsequence of $x_n$ has a further subsequence which converges to $x$.Then the sequence $x_n$ converges to $x$.

Is the following is true? Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$. I ...
20
votes
2answers
1k views

If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
18
votes
1answer
10k views

Continuous function on a compact metric space is uniformly continuous

I am struggling with this question: Prove or give a counterexample: If $f$ is a continuous function on a compact subset $Y$ of a metric space $X$, then $f$ is uniformly continuous on $Y$. ...
16
votes
2answers
6k views

A and B disjoint, A compact, and B closed implies there is positive distance between both sets

Claim: Let $X$ be a metric space. If $A,B\in X$ are disjoint, if A is compact, and if B is closed, then $\exists \delta>0: |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$. Proof. ...
3
votes
1answer
320 views

How to show that the spherical metric satisfies the triangle inequality?

For $x,y\in \mathbb R^n$ define $$d(x,y)={\|x-y\| \over \sqrt{1+\|x\|^2} \sqrt{1+\|y\|^2}}$$ Here $\|x\|$ is the euclidean norm of a vector. How to prove that $d$ (the spherical metric) is indeed a ...
27
votes
4answers
4k views

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric ...
13
votes
2answers
2k views

if every continuous function attains its maximum then the (metric) space is compact

Suppose $(M,d)$ a metric space. I want to show that if every continuous real-valued function on $M$ attains a maximum, then the space must be compact. I was trying to do this by assuming $M$ ...
15
votes
2answers
1k views

Isometry in compact metric spaces

Why is the following true? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
6
votes
2answers
875 views

If $d$ is a metric, then $d/(1+d)$ is also a metric

Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$ Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$ ...
6
votes
4answers
2k views

Proof of the Lebesgue number lemma

I want to prove the Lebesgue number lemma: Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, there exists $\delta \gt 0$ such that for each subset of $X$ ...
5
votes
3answers
939 views

preservation of completeness under homeomorphism

Does homeomorphic metric spaces preserves completeness?I mean two metric space which are homeomorphic and one of them is complete$\Rightarrow$ another one is also complete?
2
votes
1answer
147 views

If $T^n$ is $q$-contractive, $T$ exactly has one fixed point

Consider a complete metric space $(X,d)$ and $T\colon X\to X$. Suppose there exists $n\in\mathbb{N}$ such that the n-th power of $T$ is $q$-contractive. Show that then $T$ has exactly one ...
8
votes
1answer
3k views

Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$ such that $\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$. Prove A have a unique fixed point in $K$. The ...
11
votes
4answers
4k views

Equivalent metrics determine the same topology

Suppose that there are given two distance functions $d(x,y)$ and $d_1 (x,y)$ on the same space $S$. They are said to be equivalent if they determine the same open sets. Show that $d$ and $d_1$ are ...
12
votes
1answer
932 views

Why are metric spaces non-empty?

I'm just second-marking some exam scripts, and I wanted to leap on a question and made the following pedantic remark concerning the model answers: "if the metric space is empty then this proof doesn't ...
10
votes
4answers
6k views

An open ball is an open set

Prove that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open. My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an ...
8
votes
4answers
2k views

Finding a homeomorphism $\mathbb{R} \times S^1 \to \mathbb{R}^2 \setminus \{(0,0)\}$

Are there any specific 'tricks' or 'techniques' in finding homeomorphisms between topological or metric spaces? I'm trying to construct a homeomorphism between $\mathbb{R} \times S^1 \to \mathbb{R}^2 ...
3
votes
2answers
159 views

Irrational P-adics

$\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete. I just want to be sure, are there p-adic rationals? If there are P-adic ...
12
votes
3answers
2k views

Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space

I have to show that the real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space. Certainly, I have to search for a Cauchy sequence of real numbers ...
14
votes
2answers
243 views

What operations is a metric closed under?

Suppose $X$ is a set with a metric $d: X \times X \rightarrow \mathbb{R}$. What "operations" on $d$ will yield a metric in return? By this I mean a wide variety of things. For example, what functions ...
8
votes
1answer
328 views

If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected

I have an exercise found on a list but I didn't know how to proceed. Please, any tips? Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of ...
4
votes
1answer
491 views

which of the following metric spaces are complete?

Which of the following metric spaces are complete? $X_1=(0,1), d(x,y)=|\tan x-\tan y|$ $X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$ $X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$ $X_4=\mathbb{R}, ...
10
votes
3answers
1k views

Continuous extension of a uniformly continuous function from a dense subset.

I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous ...
2
votes
1answer
2k views

Showing $(C[0,1], d_1)$ is not a complete metric space

I am completely stuck on this problem: $C[0,1] = \{f: f\text{ is continuous function on } [0,1] \}$ with metric $d_1$ defined as follows: $d_1(f,g) = \int_{0}^{1} |f(x) - g(x)|dx $. Let the sequence ...
7
votes
1answer
585 views

Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
2
votes
3answers
318 views

Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric space.

Show $\rho (x,y)=\dfrac{d(x,y)}{1+d(x,y)}$ is a metric on the metric space $X$, equipped with the Euclidean metric $d$. I've already shown that the positivity $\rho(x,y)\geq 0$, the symmetry ...
1
vote
1answer
147 views

Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
0
votes
1answer
269 views

What about the continuity of these functions in the uniform topology?

Let $f$, $g$, $h \colon \mathbf{R} \to \mathbf{R}^\omega$ be defined by $$\begin{align*} f(t)&:=(t,2t,3t,\ldots),\\\\ g(t)&:=(t,t,t,\ldots),\\\\ ...
30
votes
2answers
7k views

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
40
votes
5answers
2k views

Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?

Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space? Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
10
votes
3answers
10k views

Show that in a discrete metric space, every subset is both open and closed.

I need to prove that in a discrete metric space, every subset is both open and closed. Now, I have problem imagine how this space looks like. I think it contains of all sequences containing ones and ...
7
votes
2answers
219 views

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ ; then how to prove that $d(f(x),f(y))=d(x,y) , \forall x,y \in M$ i.e. that $f$ is an ...
6
votes
1answer
217 views

Continuity of $d(x,A)$

I am doing a head-check here. I keep seeing this theorem quoted as requiring $A$ to be closed (as in Is the function distance continuous?), but I don't think that it is needed. Theorem. Let ...
5
votes
3answers
5k views

Prove that a finite union of closed sets is also closed

Let $X$ be a metric space. If $F_i \subset X$ is closed for $1 \leq i \leq n$, prove that $\bigcup_{i=1}^n F_i$ is also closed. I'm looking for a direct proof of this theorem. (I already know a ...
4
votes
4answers
2k views

Are all metric spaces topological spaces?

I think this is true but i cannot prove it. Any answer or hints are welcome. I have tried to start with $\mathbb{R}$ with euclidean metric. We may consider $\tau :=\{\emptyset,\mathbb{R}\}$ and ...
9
votes
1answer
290 views

Question(s) about uniform spaces.

I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space. It ...
6
votes
1answer
867 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
3
votes
1answer
262 views

Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$

Let $(X,d)$ be a metric space. Prove that $E$ is disconnected iff there exists two open disjoint sets $A$,$B$ in $X$ such that $E\cap A\neq\emptyset, E\cap B\neq \emptyset$ and $E\subset A\cup B$. ...
3
votes
2answers
298 views

Is every set in a separable metric space the union of a perfect set and a set that is at most countable?

I'm reading Baby Rudin and exercise 28 of chapter 2 reads "Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable." ...
5
votes
3answers
1k views

Notions of equivalent metrics

Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...
1
vote
1answer
86 views

Why does the natural quotient metric works in this case?

There have been at least two discussions in this forum about why a (pseudo) metric is defined in the quotient of a metric space by: $$ d([x],[y]) = \inf \{ d(p_{1}, q_{1}) + \ldots, d(p_{n}, q_{n}) \} ...
0
votes
2answers
301 views

Show that $d$ is a metric on $\mathbb C^n$

On $\mathbb C^n$, define $||z||=(\sum_{j=1}^{n}|z_j|^2)^\frac{1}{2}$ and for $x,z\in\mathbb C^n$ define $d(z,w)=||z-w||.$ Prove that $d$ is a metric on $\mathbb C^n$. My attempt: I need to show ...
72
votes
0answers
2k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
14
votes
6answers
784 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
3
votes
1answer
78 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...