0
votes
1answer
26 views

Why is this set closed?[metric-spaces]

I am reading a note, where part of it is this: Why is S' closed? I have tried to argument like this, but I am not able to finish the argument: Let $\{x_n\}$ be a convergent sequence in S', then ...
1
vote
2answers
204 views

How do I turn my verbal argument into something formal in [Real Analysis]? (proving every compact set is bounded)

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
0
votes
0answers
36 views

How do I turn my verbal argument into something formal in Real Analysis? [duplicate]

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
2
votes
4answers
141 views

How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...
1
vote
1answer
24 views

Lipschitz continuous set-valued map

Let $X$ and $Y$ be metric spaces and $F:X\to2^Y$ be a set-valued map. Suppose that $2^Y$ is endowed with the Hausdorff metric. I wonder about sufficient conditions on $F$ that ensure this map is ...
1
vote
2answers
57 views

We know that a Compact set is closed. However a finite discreet set is compact but not closed (contradicting the theorem?)

We know that a Compact set is closed. We also know that a finite discreet set is compact (as every cover has a finite sub cover). However a finite discreet set is not closed (contradicting the ...
0
votes
1answer
20 views

metric-spaces closure

The line in my writing symbolizes the closure, where we add all the boundary points. Is it true that: $\overline{A \cap B}=\overline{A}\cap {B}$? If B is closed, but we do not know if or if not A ...
2
votes
1answer
38 views

Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1 $)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1 $) if they induce equivalent metrics. At first I thought this ...
0
votes
0answers
16 views

Proof verification related to the discrete metric

Can someone please verify my proof? Let $X_1$ be a set and let $d_1$ be the discrete metric on $X_1$. (a) Prove that every subset of $(X_1, d_1)$ is open. (b) Prove that if $(X_2, d_2)$ ...
0
votes
1answer
23 views

Condition for Lipschitz functions and set inclusion

Let $(X,d)$ be a pseudometric space, and for each $A\subseteq X$ and $\varepsilon \geq 0$ define $$ A^\varepsilon := \{x\in X:\exists a\in A \text{ s.t. }d(x,a)\leq \varepsilon\}. $$ Is this set ...
3
votes
1answer
50 views

Qualifier problem: Completeness of Metric Spaces

I am working on old qualifier problems as a review, and I came across this one: Suppose there exists a continuous surjection $f:X_1 \mapsto X_2$, where $(X_1,d_1),(X_2,d_2)$ are metric spaces, such ...
0
votes
2answers
44 views

if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.
1
vote
1answer
43 views

How to check whether this function is continuous or not..?

Let A and B be two disjoint closed sets of any Metric space X.I have to construct a continuous function such that $f(x):= 0$ if x belongs to $A$ $f(x) = 1$ if x belongs to $B$ My idea is to use the ...
1
vote
0answers
40 views

Characterization of continuity by subsequences.

I have a difficulty trying to prove the following proposition. Any help would be greatly appreciated. $\textbf{Prop.}$ Let $(X,d_1)$ and $(Y,d_2)$ be two metric spaces. A function $f:X\rightarrow Y$ ...
0
votes
0answers
33 views

Verifying the triangle inequality for a metric for hyperbolic space

I read that the formula $d(x,y)=\mathrm{arccosh}(1+\frac{2||x-y||^{2}}{(1-||x||^{2})(1-||y||^{2})})$, where $x,y$ are in the open unit ball of $\mathbb{R}^{n}$ and $||\cdot||$ denotes Euclidean norm, ...
0
votes
3answers
41 views

Which of the following sets are dense in $C[0,1]$

Which of the following sets are dense in $C[0,1]$ with respect to sup-norm topology? $1$. {$f$$\in$ $C[0,1]$ : $f$ is a polynomial } $2$. {$f$$\in$ $C[0,1]$ :$f(0)$=$0$} $3$. {$f$$\in$ $C[0,1]$ ...
0
votes
2answers
24 views

replace convergence with continuity?(metric spaces)

This question is convcerning metric-spaces. In theory we can replace continuity with convergence. That is, since continuity in a point a is equal to the statement that if $\{x_n\}$ is any sequence ...
1
vote
1answer
42 views

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz. Is the above statement true? I thought since $f$ is continuous on a compact metric space, $f$ ...
1
vote
1answer
33 views

Can I show these questions (is a set open or closed WRT metric) a faster way?

I have the metric: $$d((x,y),(a,b))=|y-b|\text{ if }a=x\text{ else }|y|+|b|+|x-a|$$ I have been asked the following questions: Is the set $\{0\}\times(0,1)$ open with respect to this metric? Is it ...
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 $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
0
votes
1answer
31 views

Is the Set of Distances Between a Finite Open Subset and a Closed Subset of a Metric Space Closed?

In order to be as clear as possible, I've taken the liberty of TeXing (Tikzing?) up the sort of image in question. Here, $\gamma$ is some path in the complex plane, the disk ...
8
votes
1answer
81 views

Complete metric subspaces of $\mathbb{Q}$

Is there a nice characterization for the complete metric subspaces of $\mathbb{Q}$ (with the usual metric)? It seems like a such a subspace must have empty interior; if it contained an open interval ...
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
33 views

Counterexample $||S+T||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}+||T||_{\mathbb{B}(E)}$

Let $S,T\in \mathbb{B}(E),\ \mathbb{B}(E)=\left\{T:E\to E:T\ linear\ bounded\right\}$ Give a countraexamples such that: (a) $||S+T||_{\mathbb{B}(E)}<||S||_{\mathbb{B}(E)}+||T||_{\mathbb{B}(E)}$ ...
1
vote
1answer
27 views

About interior of the frontier (proof-checking)

Let $M$ be a metric space, and $A \subset M$ an open set. Show that $\stackrel{o}{\widehat{\partial A}} = \emptyset$. ($\stackrel{o}{\widehat{\partial A}}$ is the interior of the frontier) I ...
1
vote
1answer
26 views

Compatible maps

Definition Let $M$ be a subset of a metric space $X$ and $T, I :M\to M$ be $M$-invariant maps. Then the pair $(T,I)$ is called compatible if $$\lim_{n\to\infty}d(ITx_n, TIx_n)=0$$ whenever ${x_n}$ is ...
5
votes
3answers
98 views

Why must an interior point of $E$ be an element of $E$?

This question takes place in a general metric space $X$. Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. This is like the ...
0
votes
0answers
37 views

All metrics on finite spaces are equiv - I'm happy with this, except for that annoying point metric.

Okay for $\mathbb{R}^2$ say, I'm quite happy that $d_1(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ and $d_2(x,y)=\max(\{|x_1-y_1|,|x_2-y_2|\})$ that the unit ball is a circle in one and a square in the other ...
0
votes
1answer
26 views

Simple openness/closedness question

I read on another thread here that the set $\{0\}$ is open in $\{0,1\}$, with $\{0, 1\}$ a subset of $\textbf{R}$. This makes sense to me b/c $\exists$ an open set in $\textbf{R}$, say, $(-1,1)$ s.t. ...
10
votes
1answer
217 views

Is this a metric?

I now that one can show that if $d$ is a metric on a vectorspace $X$ then so is $$\varrho(x,y):=\frac{d(x,y)}{1+d(x,y)}.$$ This easily follows from the fact that the function $s \mapsto \frac{s}{1+s}$ ...
1
vote
0answers
63 views

Proving that there is no norm for the space of real-valued sequences making it a complete metric space.

Suppose I have a vector space $K$ which consists of real-valued sequences with only finitely many non-zero terms. I would like to show that there doesn't exist a norm on $K$ that would make it become ...
-1
votes
4answers
59 views

A subspace of a separable metric space is separable [closed]

Prove that a subspace of a separable metric space is separable my attempt to solution is Let $X$ be a separable metric space take $A \subset X$ i need to show that A is separable i.e show that $A$ ...
2
votes
1answer
66 views

Conditions for Metricization of Cartesian Product of Metric Spaces

Let $M_1$ and $M_2$ be metric spaces with metrics $\rho_1$ and $\rho_2$ respectively. What are some necessary and sufficient conditions on $f:\mathbb{R}_{+}^2\to\mathbb{R}_{+}$ that make ...
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 ...
7
votes
2answers
177 views

What is the completion of this space?

This question asks us to show that $\Bbb R$ with the following metric is not complete: Fix a strictly positive function $f \in L^1(\Bbb R)$, and let $d(x,y)=\left|\int_x^y f(t)dt\right|$. It's easy ...
-1
votes
1answer
56 views

Some Property of Cantor set?

Draw a Cantor set $C$ on the circle and consider the set $A$ of all chords between points of C. Prove that $A$ is compact. Is $A$ convex? The proof of first part goes as follows: As we know ...
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
vote
1answer
76 views

Topology. Why is $T^{-1}$ continuous?

Today we did this proof, but we could not finish it and our prof said that the end would be easy, but I could not finish this proof. Let $X$ be a $T_3$ space with a countable basis $B$. Then we ...
1
vote
1answer
32 views

a decreasing sequence of convex domains

Let $\Omega_1 \supset \Omega_n \supset\cdots$ a decreasing sequence of bounded, convex and open sets in $R^n$. Define $\Omega = \operatorname{int} \left(\overline{\bigcap \Omega_n}\right)$ and supose ...
3
votes
2answers
140 views

Compact metric connected space

If I have a compact metric space $X$ such that for all $a,b \in X$, there are points $a:=x_1,...x_n=:b$ such that $d(x_i,x_{i+1})< \varepsilon$, then this space is connected. Somehow, I don't see ...
1
vote
3answers
195 views

Can't prove statement jumped over in book proof, I actually think it might be wrong. But it is used throughout the book

I wish to prove that the metric $d(x,y)=\sum^\infty_{n=1}\frac{1}{2^n}\frac{|x_n-y_i|}{1+|x_n-y_n|}$ This is a metric on infinite sequences, x,y and z are sequences with terms $x_i$ and so forth. ...
0
votes
1answer
18 views

Hausdorff is a metric outer measure

I am new to measure& hausdorff measure, when looking at the proof of this property, I have a question : Given $E_1,E_2 \subset X,X$ is a metric space, we want to prove that if ...
1
vote
2answers
92 views

If$ (X,d)$ is a metric space, is $(X,d^2)$ a metric space?

If $(X,d)$ is a metric space, is $(X,d^2)$ a metric space or not? Thanks a lot.
0
votes
1answer
34 views

Proof strategy - Borel $\sigma-$fields

How does one go about proving the following: Every open set $A$ in the topological space $(\mathbb{R}^d,\|\cdot\|)$ (with the norm topology) is the union of all the open balls $B_\epsilon(q)$ whose ...
2
votes
3answers
38 views

On Contractive mapping theorem

I am trying to solve one of the problem in the book "Berkeley Problems in Mathematics": Define a sequence of positive numbers as follows. Let $x_0>0$ be any positive number, and let ...