0
votes
0answers
19 views

Question about Boundary points of the sets in metric space

Let A be a metric spaces. Prove the following properties: The boundary of $A$ equals $A'-A$ The boundary of $A$ is the closed set. $A$ is closed if and only if it contains its boundary. Where ...
0
votes
2answers
35 views

Number of open sets in a metric space

I have got the following question which I could not solve: can a metric space have exactly 36 open sets? I believe if the metric space is finie, then it has to be discrete and so the number of open ...
2
votes
1answer
52 views

For compact $K$ and open $U \supseteq K$, there exists $\varepsilon>0$ such that $B(K,\varepsilon) \subseteq U$

Let $X$ be a metric space. Let $K$ be a compact subset of $X$ and $U$ an open subset of $X$ containing $K$. I strongly believe and want to prove that there exists $\varepsilon>0$ such that ...
4
votes
2answers
75 views

Prove that $[0,1]$ is not isometric to $[0,2]$.

Prove that $[0,1]$ is not isometric to $[0,2]$. Suppose there is an isometry $f:[0,1]\to[0,2]$. Since f is continuous and surjective, the only values for $f(0)$ and $f(1)$ are $f(0)=0$ and ...
0
votes
1answer
57 views

Subset of infinite connected set

How to proove that infinite connected set has got proper infinite connected subset?
0
votes
2answers
71 views

Question on two metric spaces properties

Question: Let $X$ be a set and let $d_1$ and $d_2$ be two metrics on $X$. Assume that there exists a constant $C > 0$ such that $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
4
votes
1answer
42 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
0
votes
1answer
48 views

Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
0
votes
1answer
21 views

Continuous function in metric spaces

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. $f(x): (P,\rho) → (\mathbb R,\rho_1): 0$ if $x∈[-1,1]$ or $1$ if $x∈\mathbb R/[-1,1]$. $\rho_1 (x,y) = \sum_{k=1}^\infty |x_k-y_k|$. ...
0
votes
1answer
34 views

Non-constant Cauchy sequence

I need to find an example of non-constant Cauchy sequence in $\mathbb E^2$. The metric in question is $\rho_2$, so Cauchy sequence would be sequence for which following is true: $\sqrt {(x_m - y_m)^2 ...
0
votes
1answer
21 views

Proving that $b \in \overline{A}$ if and only if $\rho(b,A) = 0$

I need some help with this problem: Let be $(X,\rho)$ a metric space, $A \subseteq X$ and $b \in X$. The distance from $A$ to $b$ is defined as $\rho(b,A) = \inf\,\{ \rho(b,a) : a \in A \}$. Prove ...
3
votes
1answer
108 views

About equivalent norms

Consider $E$ the space of the functions $f: [0,1] \to \mathbb{R}$ such that $f(0) = 0$ and $f$ satisfies a Lipschitz condition. We define two norms: $$\|f\| = \sup_{x \in [0,1]} |f(x)|$$ and ...
0
votes
1answer
25 views

Distance in metric space p_{1}

I need to evaluate distance of point [6,6] and circle $x^2 + y^2 = 25$ in metric space $p_{1}(x,y) = ∑|x_k-y_k|$ (sum metric). I know that I need to count $inf(p_{1}([6,6],X), X $ are points from ...
1
vote
3answers
46 views

Proving a metric with absolute value [duplicate]

I need to prove that function $\mathbb R × \mathbb R → \mathbb R $ : $f(x,y) = \frac{|x-y|}{1 + |x-y|}$ is a metric on $\mathbb R$. First two axioms are trivial; it's the triangle inequality which is ...
-1
votes
2answers
43 views

A Fundamental Property of Metric Spaces …

Let $(X,d)$ be a metric space and $A\subset X$ and also suppose that $G$ is open in $X$ prove the identity: $$ \overline {G\cap A}=\overline {G\cap \overline A} $$ Proposition: The intersection of ...
0
votes
0answers
30 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
-1
votes
1answer
24 views

Some Topological Properties of Starlike Sets!

A subset $E$ of $\mathbb R^n$ is starlike if it contains a point $p_0$ (called a center for $E$) such that for each $q\in E$, the segment between $p_0$ and $q$ lies in $E$. For more information please ...
0
votes
1answer
27 views

Possible forms of open balls

Consider $X= ( \Bbb Q \cap [ 0,3] , d_E)$The question is as such: "Describe the possible forms that an open ball can take in $X = (\Bbb Q ∩ [0, 3], d_E )$." I don't understand this means exactly. ...
1
vote
0answers
16 views

Creating a metric from a pseudometric

Given the following definition of a pseudo-metric on the set $X$ : A pseudo-metric on the set $X$ is a map $d:X \times X \to \Bbb R^+$ such that for all $x ,y \text{ and } z \in X :$ (PM1) $x=y ...
0
votes
1answer
36 views

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact.

Let $M$ be a bounded subset of the space $C_{[a,b]}$. Prove that the set of all functions $F(x)=\int^{x}_{a}f(t)dt$ with $f\in{M}$ compact. Some helpful definitions: bounded - A subset $S$ of a ...
1
vote
1answer
25 views

A complex metric

Given the following definition $d(z , w) = \begin{cases}0 & z=w \\ |z|+ |w| & z\neq w \end{cases}$ I have to prove that $d(z,w)= 0\Rightarrow z = w$ Which is in part of checking that $d$ is ...
2
votes
1answer
50 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
1
vote
1answer
24 views

find a sequence of closed connected subsets $V_n$ of $\mathbb R^2$ s.t. $V_n\supseteq V_{n+1}$ and $\cap^{\infty}_{i=1}V_i$ is not connected

Find an example of a sequence $V_n$ of closed and connected subsets of the Euclidean plane satisfying $V_n \supseteq V_{n+1}$ such that $\cap_{i=1}^{\infty}V_i$ is not connected. Normally I would ...
0
votes
1answer
23 views

Closure of interior proper subset of interior of closure

$\newcommand{\intr}{\mathrm{int}}$ I need to give an example of a metric space $(X,d)$ and $A ⊆ X$ so that $\overline{\intr(A)} ⊂ \intr (\overline{A})$, where $\overline{B}$ refers to the closure of ...
-1
votes
1answer
34 views

proving the equivalence of to metrics

Any hints as to how I can prove that $(\mathbb{R}^n,d_\infty)$ and $(\mathbb{R}^n,d_T)$ are topologically equivalent. Where $$d_\infty = \sup\{|x_i-y_i|, |x_2-y_2|,..,|x_i-y_i|\} $$ and $$d_T = ...
0
votes
1answer
45 views

Proving a basis exists

How would I show that there exists some set of open balls with rational radius and rational centre such that they are a subset of the reals.That is, $\exists p,q\in \mathbb{Q} $ and $ r,x \in ...
1
vote
1answer
26 views

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and ...
3
votes
1answer
74 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
0
votes
2answers
54 views

A open subset of $\Bbb R$

Given the definitions in Open Subsets of open sets I need to prove that $\{x \in \Bbb R : |x|>2\}$ is open in $(\Bbb R , d_E)$ This seems to be true, however I don't know how to prove it without ...
0
votes
1answer
26 views

Possible metric space

$d_m$ is defined on $\Bbb R^2$ as such: $d_m(x,y) = max \lbrace|x_1 - y_1| , |x_2 -y_2| \rbrace $ where $x=(x_1,x_2) , y = (y_1 ,y_2)$ Which I have the task of proving whether or not the above s a ...
0
votes
0answers
50 views

Prove that $(X,d_\infty)$ is complete.

Consider the set $X = \{f: I \to \mathbb{R}^2$:$f$ is continuous} (here $I = [0,1]$, prove that $(X,d_\infty)$ is complete where $d_\infty(f,g) =\sup_{x \in I}|(f(x)-g(x)|$. The hint we are given is ...
2
votes
1answer
73 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
3
votes
0answers
57 views

Proof about sequences of functions.

Is this proof correct? If $\{f_{n}\}$ is a sequence of functions in $C(X,Y)$, $X$ compact, $Y$ complete, and the sequence converges, to $f$, then $K=(\bigcup\{f_n\})\cup \{f\}$ is closed. Proof. ...
3
votes
2answers
63 views

The distance between an element and a subset of a metric space.

I got stuck on an assignment. Can you help me to solve this? Let $(X,d)$ be a metric space, and let $C$ be a subst. Define the function: $$ f \quad : \quad X \longrightarrow \mathbb{R} \quad : ...
0
votes
0answers
36 views

Show that $A$ is closed in $X$ and $f(A)$ is not closed in $Y$.

Let $X=[0,1)$ with the metric $d(x,y)=|x-y|$, and $Y=\mathbb{R}^2$ with the Euclidean metric. Define the mapping $f:X\rightarrow{Y}$ by $f(t)=(cos(2 \pi t + \frac{\pi}{2}), sin(2 \pi t + ...
0
votes
0answers
44 views

Show the following is Cauchy:

I am trying to prove that the Euclidean Norm/inner product on C([0,1]) does not give rise to a complete metric space. To do this I am trying to find a Cauchy Sequence which does not converge in ...
0
votes
1answer
45 views

(i) Show that T is continuous on $(X,d)$. (ii) Show that T is continuous on $(X,d_{2})$.

Let $K(t,s)$ be a continuous function on $[0,1]\times{[0,1]}$. Let $X=C[0,1]$ be the set of continuous functions defined on the interval $[0,1]$. Define the mapping $T:X\rightarrow{X}$ by: for every ...
3
votes
0answers
66 views

Conditions to make a function a metric on $\mathbb{R}$?

Let $f:\mathbb{R}\rightarrow \mathbb{R}$. What conditions ensure that $d(x,y)=|f(x)-f(y)|$ defines a metric on $\mathbb{R}$ Let $g:[0,\infty) \to \mathbb{R}$. What conditions on $g$ ensure that ...
1
vote
1answer
34 views

Is this claim harder to prove for arbitrary metric spaces than for the reals?

Let $X$ and $Y$ be metric spaces. Define the distance between functions $f, g$ from $X$ to $Y$ as $$d(f, g) = \sup_{x \in X} \frac{d(f(x), g(x))}{1+d(f(x), g(x))}$$ Is it true that if $f_n:X \to Y$ ...
2
votes
0answers
92 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 $E\subset A\cup B$. ...
0
votes
1answer
87 views

Prove that $\exists\ C$ such that $T: C \to C$ with $C=\overline{\text{conv}}T(C)$

I have a problem: Let $K$ be a nonempty, closed, bounded and convex subset of reflective Banach space $X$. Suppose that $$T:K \to K$$ is nonexpansive. Prove that $\exists\ C$ such that ...
2
votes
0answers
33 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
2
votes
2answers
55 views

Open sets are $\mu*$-measurable in a metric space with a given condition

I have a homework problem that I'm very stuck on. The problem statement is as follows: "Suppose that $X$ is a metric space, and that for any sets $E,F \subseteq X$, if dist$(E,F) > 0$ then ...
2
votes
1answer
89 views

if a subsequence of cauchy sequence converges, then the whole sequence converges.

Let $(X,d)$ be a metric space, and say $(x_n)$ is a cauchy sequence such that it has a convergent subsequence $(x_{n_k})$ that converges to $x$. We show $x_n \to x$. Let $\epsilon > 0$. Take $N ...
0
votes
2answers
84 views

Prove that $|x_1-y_1|+|x_2-y_2|$ is a metric

I have an exercise that states: a) Prove that for $0<p<1$, $d_1(x,y) =(|x_1-y_1|^p+|x_2-y_2|^p)^{1/p}$ is not a metric on $\mathbb{R}^2$ and b) Prove that for $0<p<1$, $d_2(x,y) = ...
2
votes
2answers
330 views

distance between sets in a metric space

I was given this innocent looking homework question. Given two nonempty sets $A,B \subseteq X$ where $(X,d)$ is a metric space. Show that $\mathrm{dist}(A,B) = \inf \{d(x,y) \mid x \in A, ...
0
votes
0answers
51 views

Pyramid Question

Let us suppose we have a right pyramid,that is a pyramid with all edges equal to each other and a height which goes to the center of the geometric shape that serves as it's base. We than also know ...
2
votes
0answers
81 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
2
votes
0answers
76 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
0
votes
2answers
42 views

Given a metric space $(X,\rho)$, prove that $|\rho(x,z)-\rho(y,u)|\leq{\rho(x,y)+\rho(z,u)}$ for $x, y, z, u\in{X}$.

Obviously it is true, but I'm not sure how to prove it. I'm considering the quadrilateral inequality but so far it has not been helpful. Can anyone give me direction on how to verify ...