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)

1
vote
2answers
31 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
0
votes
2answers
23 views

Show restriction map is a contraction/lipschitz mapping

For $C[a,b]$ (set of all continuous real valued functions), define $d(f,g) = \int^{b}_{a}|f(x)-g(x)|dx$ If $[c,d]$ is a subinterval of $[a,b]$ and the mapping $r:C[a,b] \rightarrow C[c,d]$ ...
0
votes
1answer
35 views

Show that a set is not open

Suppose $U_1$ and $U_2$ are both nonempty subsets of $\mathbb R$ such that $U_1 \cap U_2 =\emptyset $ and $U_1\cup U_2 = \mathbb R.$ Consider points $p \in U_1\ \text{and}\ q \in U_2.$ Without loss ...
1
vote
1answer
42 views

How to prove the triangle inequality for this distance?

I'm studying a proof in 'An Introduction to Metric Spaces and Fixed Point Theory' (M. Khamsi, W. Kirk) that shows the equivalence of injectiveness and hyperconvexity for metric spaces. I stumbled over ...
1
vote
1answer
31 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
0
votes
0answers
32 views

Prove that this infinite sum involving metrics is also a metric

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho_i: X\times X\to \Bbb R^+$ with ...
0
votes
2answers
44 views

Prove that this is a metric space

The following is a question from a previous assignment that I was unable to complete. Any assistance on how to complete this would be appreciated. Let $\varrho: X\times X\to \Bbb R^+$ be a metric on ...
5
votes
3answers
61 views

Show that $d(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric on $\mathbb{R}$ (triangle inequality) [duplicate]

Prove that $d(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric on $\mathbb{R}$. Definition. A function $d:E \times E \mapsto [0, \infty)$ is called a metric iff whenever $x,y,z \in E$, $d(x,y) = 0$ if ...
0
votes
1answer
43 views

Compactification of a straight line

Like in the case of mapping a infinite-plane to a sphere (Riemann Sphere), I can understand, that I can map the infinite line ($-\infty,\infty$) to a circle. Secondly, I can also map a finite line ...
0
votes
0answers
15 views

Question on p-adic norms and metric spaces

I was given this assignment question: I am given two distinct primes $ p \neq q $ and I am asked to produce first of all a set closed with respect to p-adic norm but not closed under q-adic norm. The ...
3
votes
0answers
32 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\frac1x-\frac1y|$ inducing same topology on $[1,\infty)$?

My textbook says the following: Let $X=[1,\infty)$ and define $d_1(x,y)=|x-y|$ and $d_2(x,y)=|\frac1x-\frac1y|$ . Then $d_1$ and $d_2$ are inducing the same topology, $(X,d_1)$ is complete but ...
3
votes
1answer
34 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
0
votes
1answer
39 views

Proving that two arbitrary circles are homeomorphic

I came across this practice question, which seems rather simple - but I am wondering if I am not understanding something completely. If I were to define an explicit homeomorphism to demonstrate that ...
0
votes
1answer
55 views

What is the interior of a single point in a metric space?

Let $(X,d)$ be a metric space. We know that if $x \in X$ , then $Cl(\{x\})=\{x\}$, which implies that $\{x\}$ is closed. However if that's the case, what would the interior of $\{x\}$ be? I was ...
0
votes
1answer
24 views

Metric Fixed Point Theory

I am learning Metric Fixed Point Theory by Mohammed A Khamsi and William A Kirk. I need help in understanding a step in the proof of the following theorem(Chapter 3, Theorem 3.2, Page No. 43): ...
0
votes
2answers
35 views

$\{\infty\}$ open in $\mathbb N\cup\{\infty\}$ with $d(a,b)=|\arctan a-\arctan b|$?

Let $X=\mathbb N\cup\{+\infty\}$. I want to find two metrices inducing different topologies. Let $d_1$ be the discrete metric then all subsets of $X$ are open. (in particular $\{+\infty\}$) But now ...
0
votes
1answer
19 views

Give a example of a sequence of continuous functions which do not form a Cauchy sequence

As an example that not every Cauchy sequence in $(M,d)$ is converging in $M$ the following examples are given: Consider $(\mathbb{Q},d_{\text{eucl}})$ and a sequence $q_n \in \mathbb{Q}\to ...
2
votes
1answer
90 views

Showing a function $f:X \rightarrow Y$ is continuous

I am working through some practice questions, and I am not sure if I am on the right track with this one: Let $X = \cup_{n≥1}A_n$, be a topological space and assume that a map f : X → Y is such ...
1
vote
2answers
68 views

Metric spaces are completely normal

Given a metric space $(X, k)$ with $Y, Z\subset X$ and $\operatorname{cl}(Y)\cap Z = \emptyset$, $\operatorname{cl}(Z)\cap Y = \emptyset$, prove that there are open sets $M, N$ such that $Y\subset ...
4
votes
3answers
59 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
1
vote
2answers
47 views

If $E = \bigcup_{n = 1} ^ \infty \left(\frac{1}{n+1}, \frac{1}{n}\right) $. Show $\bar E = [0,1]$ is the closure of $E$

If $E = \bigcup_{n = 1} ^ \infty \left(\frac{1}{n+1}, \frac{1}{n}\right) $. Show $\bar E = [0,1]$ is the closure of $E$ Attempt: Since $ (\frac{1}{n+1}, \frac{1}{n}) $ is a subset of [0,1], so $E ...
0
votes
1answer
25 views

How do you prove that a metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property below?

A metric space $X$ is separable if and only if $X$ has a countable subset $Y$ with property: for $\epsilon > 0$ and every $x \in X$, there is a $y \in Y$ such that $d(x, y) < \epsilon$.
2
votes
3answers
105 views

Open Sets in $\mathbb{R}$

I was wondering what the general form of an open set is in the real numbers. Is it just an interval of the form $(a,b)$; $a,b \in \mathbb{R}$.
1
vote
1answer
31 views

If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$.

If $E$ is closed and bounded in a metric space $X$ and $f: E → R$ is continous on $E$ , prove that $f$ is bounded on $E$. and suppose that $X$ satisfy the Bolzano Weierstrass Property attempt: ...
2
votes
0answers
41 views

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed.

The projection map $\pi_j : \Bbb R^n \to \Bbb R$ given by $\pi_j(x) = x_j$ is open but not closed. I have found an example for the map not to be closed. But unable to prove that it is open. Please ...
8
votes
1answer
96 views

Characterization of the circle within metric spaces

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the ...
1
vote
1answer
29 views

Doubt related to the extended real line and distance/metric

I am studying real analysis and I am being introduced to functions that take values on the extended real line, I have a fundamental doubt about this so I'll give an example to illustrate my confusion: ...
2
votes
2answers
21 views

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open.

If $g: Y \to Z$ is a continuous injection, then a map $f : X \to Y$ is open if $g\circ f$ is open. To show the map $f : X \to Y$ is open, we first take any open subset $U$ from $X$ and then show that ...
2
votes
0answers
32 views

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
0
votes
1answer
27 views

Sequence characterization of bounded sets

If $M$ is an arbitrary metric space, the following holds: $A\subseteq M$ is totally bounded $\Leftrightarrow$ Each sequence in $A$ contains a Cauchy subsequence. Additionally, for ...
1
vote
1answer
25 views

Is $\overline{\mathbb{R}}^+$ a compact Polish space

if $X$ is defined by $$X= [0,+\infty)\cup\{+\infty\}$$ is endowed with the metric $$d_X(x,y) = |\arctan(x) - \arctan(y)|$$ Is it true that the metric space $(X,d_X)$ meets the following properties? ...
0
votes
1answer
25 views

Alternate definition limit

The definition of the limit of a sequence is: $L=\lim\limits_{n\to\infty}f(n)\Leftrightarrow\forall\epsilon>0:\exists N\in\mathbb{N}:\forall n\in\mathbb{N}:\left(n>N\Rightarrow ...
1
vote
0answers
30 views

Name of the metric: $d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$

What is the name of the metric: $$d(f,g)=\max \limits_{a\leq x \leq b} |f(x)-g(x)|$$ Where $f,g\in X$ where $X$ is the space of all continuous functions. I can't find any documentation on this ...
1
vote
1answer
19 views

Checking my understanding of the Interior of these intervals

Let $[a,b]$ be any finite closed interval. (i) $\text{Int}_{[a,b]}(a,b]$ Am I correct to say that the interior of this set is $[a,b]$? Since the interior of a set are all the points in the set in ...
0
votes
1answer
31 views

An example of a dense and co-dense set in a metric space with countable derived set

Let $(X,d)$ be a metric space and $A\subset{X}$ such that $A$ and $A^c$ are both dense in $X$. Show that it is not necessary that $A^\prime$ be uncountable. And prove $(A^\prime)^\prime=A^\prime$. ...
0
votes
2answers
25 views

If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$.

Let $(X,d)$ be a metric space and $A,B \subset X$. If $A$ is connected , $B$ is open and closed and $A \cap B$ is non empty then $A \subset B$. I tried it with proving a contradiction if we first ...
0
votes
0answers
22 views

Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated.

A point $a$ in a metric space $X$ is said to be isolated if and only if $r> 0$ so small that $B_r(a)$ = {$a$} Show that a point $ a \in X$ is not a cluster point of $X$ if $a$ is isolated. proof: ...
0
votes
1answer
32 views

A complete subset of a metric space is closed?

Supposing $A$ is a subset of a metric space $S$, it is simple enough to show that if $S$ is complete and $A$ is closed, that $A$ is complete. However, without being given that $S$ is complete, what ...
-1
votes
0answers
20 views

Question about contraction mapping (an easy example) [closed]

Is following map a contraction on the real line? $$ f(x)=x+\frac{1}{1+e^{x}}. $$ If yes then what is the contraction coefficient?
3
votes
1answer
51 views

Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
-1
votes
0answers
25 views

How can I find the closure of $P[a,b]$ [closed]

Let $P[a,b]$ the space of all polynomials on the interval $[a,b]$ clearly $P[a,b]$ is a subspace of $C[a,b]$ but how can find the closure of $P[a,b]$ , In special case $[0,1]$ .
1
vote
1answer
19 views

Let $X$ be the union of axes is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$?

Let $X$ be the union of axes given by $xy = 0$ in $\Bbb R^2$ . Is it homeomorphic to a line, a circle, a parabola or the rectangular hyperbola $xy = 1$? If we remove the origin from the union of axes ...
0
votes
1answer
14 views

Evenly Spaced Integer Topology is Metrizable

Fustenborg's proof uses an evenly spaced integer topology on $\mathbb Z$ which declares that a basis of open sets as those of the form $a + b \mathbb Z$ (i.e. arithmetic progressions). I'm interested ...
1
vote
2answers
41 views

Is there a metric proof for the Caristi theorem?

I'm writing a paper about hyperconvexity in metric spaces and came across Caristi's theorem: Let $(X, d)$ be a complete metric space and $\phi\colon X \to \mathbb{R}^+$ a continuous function. An ...
1
vote
1answer
32 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
0
votes
1answer
24 views

How to find an open ball for a metric space?

I don't understand the process to find the open ball. I understand the definition and I understand that for B(0, delta), I need to substitute x as 0. After this stage, I don't understand where to go ...
1
vote
2answers
33 views

Help needed in a proof to show path connected-ness of a special kind of set in real Euclidean space

Let $A$ be a connected subset of $\mathbb R^n$ and $\epsilon >0 $ , consider $V:=V(\epsilon , A):=\{x \in \mathbb R^n : d_A(x)< \epsilon \}$ , I need to show that $V$ is path connected , so we ...
2
votes
3answers
51 views

Is every open cover of $\{(x,y) \in \mathbb R^2 : x^2+y^2=1\}$ also an open cover of some $\delta$ annulus around the unit circle?

Let $\{U_\alpha\}$ be an open cover of $\{x \in \mathbb R^n:\|x\|=1 \}$ , $n \ge 2$ , then does there exist $\delta >0$ such that $\{U_\alpha\}$ is also an open cover of $\{ x \in \mathbb R^n : ...
1
vote
1answer
28 views

$f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function.

Let $U$ be an open connected subset of $\Bbb R^n$ and $f : U \to \Bbb R$ be a differentiable function such that $D f (p) = 0$ for all $p \in U$. Then $f$ is a constant function. I am facing ...
2
votes
0answers
31 views

Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to ...