Tagged Questions
0
votes
1answer
29 views
Construction of bijective map $f:X\mapsto \mathbb{R}$
If $(X,d)$ is a metric space. Is it possible to construct a bijective continuous map $f(X,d)\mapsto \mathbb{R}$. I think it is not possible.Could any one help me to give me hints.
0
votes
1answer
55 views
A question on the compact subset of $[0,1]$
Let $S=\{K \subseteq [0,1]: K \text{ is compact and uncountable } \}$. How to see that $|S|=\mathfrak c$?
Thanks for your help.
1
vote
2answers
43 views
Finding a bounded, non-compact set of functions $f:[0,1]\to\Bbb R $
Consider the metric space $(X, d)$ given by $$X = \{\text{all continuous functions}\,f:[0,1]\to\Bbb R\}$$ with $$d(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|.$$ Find with proof a set $A \subseteq X$ with ...
1
vote
2answers
37 views
Is this set closed? (Finding the limit points of a set)
Prove that this set is closed:
$$ \left\{ \left( (x, y) \right) : \Re^2 : \sin(x^2 + 4xy) = x + \cos y \right\} \in (\Re^2, d_{\Re^2}) $$
I've missed a few days in class, and have apparently ...
2
votes
1answer
75 views
Continuity in metric space, TRUE or FALSE?
Let $(X,d)$ and $(Y,e)$ be metric spaces , and let $f: X \to Y$ be a function.
True or false ? Give a proof or a counterexample as appropriate.
$(a)$ If $d$ is the discrete metric on ...
1
vote
1answer
225 views
Prove that if $Z\subseteq Y$, then $(g\circ f)^{-1}(Z)=f^{-1}(g^{-1}(Z)).$
Let $W ,X$ and $Y$ be three sets and let $f :W \to X$ and $g: X \to Y$ be two functions. Consider the composition $g \circ f: W \to Y $ which, as usual , is defined bt $(g\circ f)(w)=g(f(w))$ for ...
0
votes
1answer
39 views
Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]
Where $ X $ is connected. Does it imply $ f $ to be continuous?
1
vote
2answers
50 views
Let $ f:(X, d)\mapsto (X, d ) $ be a mapping on compact metric space with $ d (f (x), f (y))<d (x,y) $for $ x\ne y $
I prove that $ f $ has a fixed point. My question is whether the point is unique and the mapping $ f $ is continuous.
3
votes
1answer
54 views
How to show that a continuous map on a compact metric space must fix some non-empty set.
Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$
I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
1
vote
1answer
24 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
1
vote
2answers
38 views
If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.
A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$
Let $(X,d)$ be a metric space and let ...
2
votes
1answer
49 views
Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous?
Where $ (X, d) $ is a metric space. I want to prove it using sequential criteria. How do I tackle it?
0
votes
2answers
60 views
find open balls $B_1,B_2,B_3,\ldots$ so: $U=\bigcup _{n\in \Bbb N} B_n$ , where $U=\{(x,y)\in \Bbb R^2 : y\gt x\}$
In the metric space $(\Bbb R^2,d_{\Bbb R^2})$: How can I find open balls $B_1,B_2,B_3,\ldots$ so:
$U=\bigcup _{n\in \Bbb N} B_n$, where:
$U=\{(x,y)\in \Bbb R^2 : y\gt x\}$.
and why ...
0
votes
2answers
59 views
Show the $\operatorname{int}(A)$ is open.
So we want to show that the interior of any set $A$ is open.
We will denote $\operatorname{int}(A)$ as the interior of $A$ which is the set of all interior points of $A$.
I know in order to prove ...
1
vote
2answers
55 views
$U \in \tau(\Bbb R,d_\Bbb R)$ $\iff$ there are open intervals $B_1,B_2,B_3,…$ with $U= \bigcup_{n\in\Bbb N} B_n$
The rational numbers are countable: you can write $\Bbb Q =${$q_1,q_2,q_3,...$}.
Moreover,$\Bbb Q$ is dense in $(\Bbb R,d_\Bbb R)$.
Use these facts to prove for a non-empty set ...
2
votes
2answers
79 views
Show that the set of all orthogonal matrix of order $n$, $O(n)$ is a compact subset of $GL(n,\mathbb R)$
I have only concept in topology, metric space, and functional analysis. How do I tackle this. Also I want to know that is the set connected?
2
votes
1answer
82 views
Show that closed unit ball in $ l^2 $ is not compact. [duplicate]
I have a prove using the defination of compact set.But I want to prove it using sequential criteria of compactness.How is it possibe?
2
votes
1answer
83 views
determine whether or not a subset is closed or open
determine whether or not a subset is closed or open:
(a) For $X=\Bbb R^2$ and $d$ the Euclidean metric on $\Bbb R^2$:
$A_1=${$(x,y): x^2+y^2 <1$} $\cup $ {$(1,0)$}.
$A_2=${$(x,0): 0 ...
3
votes
3answers
166 views
let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
Can a finite subset of $X$ be open ?
Definitions:
a set $F\subseteq X$ is closed (in$(X,d)$) if ...
2
votes
2answers
93 views
Kernel of $p$-adic logarithm.
I'm completely clueless as to how to answer the following question:
Let $K$ be a field of characteristic zero which is complete with respect to a non-Archimedean aboslute value $|\cdot|$. Let ...
-1
votes
1answer
65 views
On the space $C[a,b]$ of continuous functions on $[a,b]$
On the space $C[a,b]$ of continuous functions on $[a,b]$
prove that $du(f,g)=\sup|f(x)-g(x)|$ is a metric.
Show that $f_{n}$ converges uniformly to $f$ on $[a,b]$ if and only if $\lim \,du(f_{n}, ...
1
vote
1answer
64 views
Space of all continuous real valued functions on $[0,1]$ with sup metric is path connected
How can I prove that the function space $\mathcal{C}[0,1]$ of all continuous real valued functions on $[0,1]$ with the sup metric is connected?
I think the sup metric is as follows:
If $f, g $ ...
1
vote
1answer
48 views
Showing $f$ is homeomorphism
Let $(X,d)$ be metric space:
Let $f$ be an isometry on $X$
Let $f(X)$ be dense is $X$.
Then how do I show that $f$ is a homeomorphism.
I have shown $f$ to be one-one. Let $f(x)=f(y)$. This ...
2
votes
2answers
68 views
How does one show that $K_1 \cap K_2 $ is compact, when $K_1 , K_2$ are compact?
Let $(X,d_X) $ be a metric space and $K_1 , K_2 $ be compact subspaces of $X$. Question: how does one show that $K = K_1 \cap K_2$ is compact?
I tried proving this by noting the following theorem: ...
0
votes
3answers
82 views
If $X$ is a metric space with infinitely many connected components, is $X$ compact?
I am having trouble with the following question. Can anyone help?
Suppose $(X ,d)$ is a metric space which has infinitely many distinct connected components. Then is $X$ compact?
Can anyone help ...
4
votes
1answer
91 views
Polynomial root (using contraction mapping principle)
I am asked to provide an iterative algorithm which would lead to finding a real root of this polynomial:
$$6x^5-x^3+6x-6=0$$
It is required to rely on the contraction mapping principle and Banach ...
2
votes
3answers
47 views
Show that the closure of a subset is bounded if the subset is bounded
Let $A$ be a subset of $X$, and let $A$ be bounded. I.e.: $\exists x_0\in X : d(x,x_0)\le K, \forall x\in A$
I want to show that $\overline{A}$, the closure of $A$ is bounded as well, but as simple ...
1
vote
3answers
65 views
Metrizability is a topological property?
How could I show that metrizability is a topological property?
Well, this means that if I have a set X that is metrizable and a homeomorphic function f from X to Y, then I need to show that Y is ...
4
votes
1answer
59 views
Prove $\operatorname{dist}(\overline{A},\overline{B}) = \operatorname{dist}(A, B)$
This is the last question on the exercise sheet and I am having real trouble formalizing my intuitions.
It should be obvious. Since the closure of a set is the set of all points in the universe with ...
4
votes
2answers
62 views
How does one show that there exists some $z \in X$ such that $f(z) = z$ under certain circumstances?
In a previous exercise, one was asked to show that the sequence $(x_n)_{n > 0}$ in $X$ (with $(X,d)$ a non-empty, complete metric space) in which we have $d(x_n,x_{n+1}) \leq \theta d(x_{n-1} , x_n ...
0
votes
1answer
48 views
If $ x $ is a limit point of a Cauchy sequence $ (x_{n})_{n \in \mathbb{N}} $, then $ (x_{n})_{n \in \mathbb{N}} $ converges to $ x $.
Define a point $ x $ in a metric space $ X $ to be a limit point of a sequence $ (x_{n})_{n \in \mathbb{N}} $ if there exists some subsequence $ (x_{n_{k}})_{k \in \mathbb{N}} $ of $ (x_{n})_{n \in ...
2
votes
3answers
41 views
If $x$ is not in $A$, a closed set in a Metric space then $d(x,A)>0$
If $A$ is a closed in a metric space $(X,d)$ with $x\notin A$, I need to show that $d(x,A)>0$.
Now assume $d(x,A)=0$ then $\exists x_n\in A $ s.t.$d(x_n,A)=0$ then there is a sequence in $A$ s.t. ...
1
vote
4answers
95 views
Show that $\rho(x,y) = |\sin(x)-\sin(y)|$ is not a metric on $\mathbb{R}$?
Show that $\rho(x,y) = |\sin(x)-\sin(y)|$ is not a metric on $\mathbb{R}$
and in what condition must be imposed on a function $f:\mathbb{R}\to\mathbb{R}$ in order for $\rho(x,y)=|f(x)-f(y)|$ to be a ...
1
vote
1answer
54 views
Metric induced Topology
The Problem: Given a metric space $(X,d)$, define a new metric $d'$ on X by $$ d'(x,y)=\frac{d(x,y)}{d(x,y)+1} $$ Is the topology induced by $d'$ the same as the topology induced by d? Prove or ...
1
vote
1answer
116 views
Prove that two normed linear spaces are equivalent as metric spaces if and only if the norms are equivalent?
We have the two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on the vectorspace V. They're equivalent if there exists a $k>0$ and $K>0$ so that $k\|\cdot\|_a\le\|\cdot\|_b\le$ K$\|\cdot\|_a$ for all ...
1
vote
1answer
38 views
Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.
Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.
My workings so far: Because this is an if and only if proof we need to show it both ways. First let's assume ...
0
votes
2answers
74 views
Distance between two sets in a metric space in different conditions
let $(X,d)$ be a metric space and let $A,B\subseteq X$. we define the distance between $A$ and $B$ as:
$$\operatorname{dist}(A,B)=\inf\{d(a,b):a \in A,b \in B\}$$
1
show that for any $x \in X$, we ...
0
votes
2answers
56 views
Does this proof make sense and correct — is it written well enough?
I'm working on a tutorial question. The question asks whether the following claim is true or false, if it is true: one is supposed to provide a proof or counter-example otherwise if it's false.
Let ...
2
votes
2answers
38 views
Confused by an argument which is used in most triangle inequality proofs in metric spaces
I'm confused by the a proof of the triangle inequality. I was supposed to prove that a function is a metric, I proved everything else except the triangle inequality.
Define $B(\mathbb{R})$ as the set ...
3
votes
2answers
77 views
Prove that $d$ is a metric
It is given that $d$ is a real-valued function on $X \times X$ which for all x, y and z in X satisfies $d(x,y)=0$ iff x=y and $$d(x,y)+d(x,z)\geq d(y,z)$$
In order to show that $d$ is a metric is ...
5
votes
1answer
90 views
In a metric space, closure is closed
In a metric space $(M,d)$, if $A$ is a subset of $M$, then $\bar A$ (closure of $A$) is closed.
My definition of $\bar A$ is $\{x\in M : \forall \varepsilon > 0, \; B(x,\varepsilon) \cap A ...
2
votes
1answer
60 views
Showing convergence in Space of Squared Summable Sequences
My Problem: Show that the sequence ${x_n}_{n\geq 1}$, where $x_n=(1,\frac{1}{2},\ldots,\frac{1}{n},0,0,\ldots)$ converges to $x=(1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots)$ in $l_2$
My ...
0
votes
1answer
40 views
Property of Integrals
The Question: Assume that we are dealing with the set of all continuous functions on $[a,b]$. What can we say about $\int_{a}^b |f(x)-g(x)|dx=0$ in terms of $f(x)$ and $g(x)$.
My Question: I am ...
0
votes
2answers
68 views
Which of the following define a metric?
Which of the following define a metric?
a. $d((x, y), (x’, y’)) = \min\{|x – x’|, |y – y’|\}$ on $\mathbb{R}^2$.
b. $d((x, y), (x’, y’)) = |x| + |y| + |x’| + |y’|$ on $\mathbb{R}^2$..
c. $D((x, y), ...
1
vote
1answer
154 views
Proof about diameter of a set
I could not prove the following question could you please help me?
Best Regards
Let $X, d(x, y)$ be a metric space. By definition, diameter of a
bounded set $A ⊂ X$ is the number $diam(A)$ = ...
2
votes
0answers
54 views
Metric Space Question
Let $(X,d)$ be a metric space and $K \subset X$. $K$ is relatively compact (or precompact) if every sequence $(x_n) \subset K$ has a Cauchy subsequence $(x_{k_n})$. Show that $K$ is relatively ...
1
vote
1answer
40 views
Metric Space - Accumulation
Let $(X,d)$ be a metric space. Let $A \subset X$ and $c \in X$. $c$ is called an accumulation point of $A$ if for every $\delta > 0$ there exists $a \in A$ such that $0 < d(a,c) < \delta$. ...
0
votes
1answer
94 views
Compact Metric Space Question
Let $(X,d)$ be nonempty compact metric space and $f : X\to X$ be a function satisfying $d(f (x), f (y)) < d(x, y)$ for all distinct pair of points $x, x \in X$. Show that $f$ have a fixed point ...
2
votes
3answers
86 views
Criteria for metric on a set
Let $X$ be a set and $d: X \times X \to X$ be a function such that $d(a,b)=0$ if and only if $a=b$.
Suppose further that $d(a,b) ≤ d(z,a)+d(z,b)$ for all $a,b,z \in X$.
Show that $d$ is a metric ...
0
votes
0answers
75 views
Continuous functions from [a,b] to $R^n$
Question from Real Analysis by Haaser and Sullivan
Let X be the set of all continuous functions from [a,b] into $R^n$ and let $d$ be defined by $$d(f,g)=max(|f(t)-g(t)|:t\in[a,b]) $$ Show that (X,d) ...
