2
votes
2answers
49 views

Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
1
vote
1answer
17 views

Is this theorem about “completion of metric space” correct?

It's well-known that there is a completion of a metric space unique upto isometry. I have tried to modify this theorem slightly and I proved this statement: Let $(X,d_X)$ be a metric space. ...
1
vote
0answers
24 views

$p$-adic metric proof

I need to prove this, Let $p$ be an odd number. It is defined the function $v_p:\mathbb{Q}\to \mathbb{Z}$ as $$v_p\left(p^n\frac{a}{b}\right)=n, \hbox{ if } \mathrm{mcd}(a,p)=\mathrm{mcd}(b,p)=1.$$ ...
1
vote
2answers
41 views

Existence of a metric space M with no continuous map from M to any other metric space

Is it possible to have a metric space M such that there is no continuous map from M to any other metric space?
1
vote
1answer
22 views

Unit ball of continuous functions is a closed set - Proof with neighborhood argument

This question is trivial if one uses sequence definition, but I want to use the usual topological definition of closed set. That is , a set is closed if its complement is open. Let $U=\{f\in ...
1
vote
1answer
40 views

Choice of Metric Gives Nice Topological Properties

I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices. Nice is defined as compact, Hausdorff, or whatever ...
3
votes
1answer
84 views

Cauchy sequence in compact metric space

Suppose $f:X\rightarrow X$ continuous function, $X$ is compact metric space with $\rho(f(x),f(y))<\rho(x,y)$ for any $x\neq y$. Let $x_n=f(x_{n-1})$, with $x_0\in X$ arbitrary. I want to show that ...
1
vote
0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
1
vote
2answers
34 views

Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
2
votes
1answer
33 views

Discontinuous function whose restriction on closed sets is continuous

Let $X$ a metric space, $\{U_i\}$ a collection of non-empty closed sets whose union is all of $X$. Give an example of a function $f:X\rightarrow \mathbb{R}$ such that the restriction $f|_{U_i}$ is ...
2
votes
3answers
101 views

Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
0
votes
1answer
28 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
211 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
146 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
28 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
59 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
40 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
24 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
61 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
46 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
44 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
43 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
35 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
44 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
38 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
34 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
84 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
29 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
99 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
28 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
218 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
65 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
70 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
110 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 ...