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.
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Contraction Mapping question
Let X be the set of continuous real valued functions defined on $[0,\frac{1}{2}]$ with the metric $d(f,g):=\sup_{x\in[0,\frac{1}{2}]} |f(x)-g(x)|$.
Define the map $\theta:X\rightarrow X$ such that ...
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1answer
110 views
Dynamics Question
Let be $T_{\beta}:[0,1]\to [0,1]$ defined by $T_{\beta}(x)=\beta x \bmod 1$ where $\beta \in (1,2).$
Questions:
$T_{\beta}$ is topologically transitive?
What about the periodic points?
...
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3answers
106 views
A characterization of open sets
Let $(M,d)$ be a metric space. Then a set $A\subset M$ is open if, and only if, $A \cap \overline X \subset \overline {A \cap X}$ for every $X\subset M$.
This is a problem from metric spaces, but ...
5
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1answer
171 views
product of hermitian and unitary matrix
Could anyone tell me how to show that, for any $g\in GL_n(\mathbb{C})$, $\exists$ $R$ a hermitian matrix with positive eigenvalues and $U$ an unitary matrix such that $g=RU$?
And (I am not sure) can ...
3
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2answers
155 views
Open Dense Subset of $M_n(\mathbb{R})$
Well, I know the fact that $GL_n(\mathbb{R})$ is open set in $M_n(\mathbb{R})$, how to show that it is dense also? Well I thought like this: If $A\in M_n(\mathbb{R})$ and If ...
3
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0answers
112 views
Sequences of Metric Spaces of Compact Subsets
Consider a complete metric space $(M, d)$ and let $F(M)$ denote the non-empty compact subsets of $M$. Then $F(M)$ is also a complete metric space under the Hausdorff distance $d_H$. Given some ...
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1answer
194 views
Proving that if $u \in A$ is an upper bound of $A$, then $u = \sup A$
Let $A\subset\mathbb{R}$ a nonempty set of real numbers bounded above and $u$ be an upper bound of $A$. Prove that if $u\in A$, then $u=\sup A$.
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1answer
259 views
Definition of Basis for the Neighborhood System
I'm trying to learn a bit about topology through independent study.
I've been using Bert Mendelson's "Introduction to Topology - 3rd edition". I'm having a lot of fun but I'm a bit confused regarding ...
6
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2answers
269 views
Pete L. Clark's Convergence Notes
I had initially sought out a better understanding of filters and nets, and a few quick google searches showed this document as highly recommended. (And they are excellent!)
I'm having a bit of ...
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1answer
60 views
$\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold
Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth.
I ...
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1answer
186 views
Uncountable product in the category of metric spaces.
I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces.
Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
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1answer
37 views
Product of Transitive Systems
Let be $M$ a topological space, and $f:M\to M$ a danymical system, i.e, a continuous map between from $M$ to $M$.
We say that a dynamical system, $f:M\to M$ is topologically transitive when, ...
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4answers
138 views
Is a metric on a metric space a bilinear form?
I've just finished a course on bilinear forms and am now starting a cause on topological spaces and was just wondering; for a metric space which is made up of a set $M$ and a metric function $d$ such ...
5
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2answers
172 views
$\epsilon$- dense subsets
Let be $M$ a compact metric space, and let $\{x_n\}$ be a dense subsequence in $M$.
We say that a set $\Lambda=\{y_1,\ldots,y_n\}$ is $\epsilon$-dense when every ball
of radius $\epsilon$ ...
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0answers
69 views
the sphere $S^n$ is a metrically homogeneous
A metrically homogeneous space is a metric space $(X,d)$ such that for all points $p$ and $q$ in $X$, there exists an isometry $f$ such that $f(p) = q$. Does the sphere $S^n$ have this property?
odd ...
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2answers
132 views
If two metrics have the same Cauchy sequences, does that imply uniform equivalence?
If two metrics $d_i$ on the same set $X$ have the same Cauchy sequences (ie. if a sequence is Cauchy for the first metric, it is also Cauchy for the other one and vice versa), can we conclude that the ...
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1answer
200 views
closed unit ball with radius 1
Is this subset compact in $l_1$ of all absolutely convergent real sequences, with the metric:$d_1(\{a_n\},\{b_n\})=\sum_{1}^{\infty}|a_n-b_n|$ closed unit ball centered at $0$ with radius $1?$ I ...
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1answer
165 views
P-adically Cauchy sequences
I am trying to do the following question
Find a p-adically Cauchy sequence which converges p-adically to $-1/6$ in $\mathbb{Z}_7$.
In general in $\mathbb{Q}_p$ what is the stronger condition, to be ...
3
votes
1answer
300 views
A closed subset of continuous functions on [0, 1]
How would one show that the set consisting of the monomials $1,x,x^2,...$ is a closed subset of the metric space $C[a,b]$ under the metric $d(a,b) = ||a-b|| =sup_{[0,1]}|a-b|$ ?
I considered its ...
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2answers
236 views
Analogue to Fixed Point Theorem for Compact metric spaces
If $\omega:H \rightarrow H$ (into) is continuous and $H$ is compact, does $\omega$ fix any of its subsets other than the empty set?
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1answer
71 views
Show that $D: C^1([a, b]) \mapsto C^0([a, b]): f \mapsto f'$ is continuous.
the problem
I have to show that a function $D: C^1([a, b]) \mapsto C^0([a, b]): f \mapsto f'$ is continuous given a metric $\| \cdot \|_{C^1([a, b])}$.
The metric $\| \cdot \|_{C^0([a, b])}$ is ...
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1answer
94 views
Equilibrium distance formula proof
Let $$d: \mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}$$ be defined by $$d(x_i,x_j)=\frac{|x_i-x_j|}{\sqrt{M(i)M(j)}},$$ where $M(i)$ represents the average distance between $x_i$ and ...
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1answer
71 views
Coordinates translation in space
First of all sorry if the title is somewhat opaque, the problem I am trying to solve is already hard to explain properly in my first language.
So, let's consider we have a plane, rectangle target in ...
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1answer
220 views
Relationship between Minkowski distance and Minkowski space
The metric induced by the p-norm:
$d((x_1,\dotsc,x_n),(y_1,\dotsc,y_n)) = \left(\sum_{i=1}^n |x_i-y_i|^p\right)^{1/p}$
is often called the Minkowski distance.
There is also Minkowski space, which ...
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1answer
67 views
Every cover covers sets with $\mathrm{diam}(A) < \lambda$ [duplicate]
Possible Duplicate:
Proof of the Lebesgue number lemma
Let $(X, d)$ be a metric space and $K \subset X$ a compact set.
Now I have to show that for all open covers $\mathcal U$, there is an ...
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2answers
311 views
The union of open balls.
Question
Show that every open subset of a metric space can be expressed as a union of open
balls.
So far I have the following:
"Let $U \subseteq X$. For each $a \in U$, choose $r_a > ...
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votes
1answer
552 views
prove that the set of rational numbers is not connected on the real line
Could someone help me through this problem?
Prove that the set of rational numbers is not connected on the real line
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1answer
255 views
Sets and limit points
Give an example of a set which
$\ \ \ $a) contains a point which is not a limit point of the set
$\ \ \ $b) contains no point which is not a limit point of the set
In part b), I think it might be ...
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1answer
254 views
Is a Metric space $(X,d)$ with $X=\{x\}$ an open set?
I've recently started to study functional analysis using "Introduction to functional Analysis" of Edwin Kreyszig. In this book there is a theorem that states that every metric space $X$ is an open ...
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1answer
411 views
Closure, boundary and interior
Describe the interior, closure and boundary of the following sets in the real line:
the set of all integers
the set of all rationals
the set of all irrationals
$(0,1)$
$[0,1]$
...
5
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1answer
119 views
Does such a subset has a nonempty interior?
Let $(a_n)_{n=1}^\infty$ be a sequence such that $0\leq a_n \leq 1$, $\sum_{n=1}^\infty a_n=1$ and let $card \{a_n: n \in \mathbb{N} \}=\infty$. Let's consider the set $$S=\{ \sum_{n\in I} a_n: I ...
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1answer
177 views
countable union of proper subspaces
In an interview I was asked to solve a question by using Baire Category Theorem (a complete metric space can not be written as union of nowhere dense subsets), the question was:
"Is the vector ...
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1answer
68 views
Bounded subspaces and diameters.
Question:
Let $X$ be a bounded metric space. Let $Y$ be a subspace of $X$. Prove that $Y$ is bounded and that $\operatorname{diam}(Y) \le \operatorname{diam}(X)$.
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3answers
277 views
preservation of completeness under homeomorphism
Does homeomorphic metric spaces preserves completeness?I mean two metric space which are homeomorphic and one of them is complete$\Rightarrow$ another one is also complete?
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2answers
146 views
Union of a connected set and its accumulation point
Let $A$ be a connected set in the metric space $(X, d)$. If $p$ is an accumulation point of $A$,then prove that $B = A \cup \{p\}$ is connected.
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2answers
182 views
Metric Space Open Sets.
Let $(X, \rho)$ be a metric. I've shown $\sigma(s,t) = \frac{\rho(s,t)}{1 + \rho(s,t)}$ is also a metric on $X$.
I'm having trouble showing that the open sets defined by the metric $\rho$ are the ...
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2answers
256 views
Integral metric.
In reading I came across the claim that the following is a metric.
For the space $X$ of all integrable functions on the interval [$0,1$] , for $f, g \in X$, the following equation defines a metric:
...
5
votes
2answers
406 views
What is the motivation of Levy-Prokhorov metric?
From Wikipedia
Let $(M, d)$ be a metric space with its Borel sigma algebra
$\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all
probability measures on the measurable space ...
1
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4answers
92 views
Functions in a metric space.
Question
Let $(X, d)$ be a metric space. For each $a \in X$, define a function $f_a\colon X \to \mathbb R$ by $f_a(x) = d(x, a), (x ∈ X)$.
Prove that for all $a, b \in X$
...
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0answers
149 views
Closed subset of closed subspace is closed in a metric space (X,d)
Is it possible for the following to hold in metric spaces?
Let (X,d) be a metric space,if A is closed in Y and Y is closed in X then A is closed in X.
If possible someone could assist me for a proof.
...
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votes
4answers
514 views
Show that for a finite metric space A, every subset is open
Let A be a finite metric space .I want to prove that every subset of A is open.
I let the set B, be any subset of A.
Since A is finite,then I know that A/B is also finite.I'm stuck here how can this ...
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1answer
130 views
Characterizing Open/Closed/compact sets in the metric space $(\mathbb{Z}^n,d)$
What is an open set in the metric space $(\mathbb{Z}^n,d)$, where $d$ is the Euclidean distance in $\mathbb{R}$?
As far as I know, in a metric space an open set $O$ is defined as follows: For each ...
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1answer
102 views
A new metric involving curves
Let $(X, d)$ be a metric space. The inner metric or length metric associated with
$d$ is the function $d_i : X \times X \to [0,\infty]$ defined by
$$d_i(x, y) := \inf L(\sigma)$$
where the infimum is ...
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3answers
66 views
Show that $(X, d_2)$ is incomplete
I have a set $X = [0, \infty)$ and two metrics:
$$ d_1(x, y) = |x-y| $$
$$ d_2(x, y) = \left| \frac{x}{1+x} - \frac{y}{1+y} \right| $$
I already showed that $d_1$ is equivalent to $d_2$. Now I have ...
2
votes
1answer
84 views
Curves and geodesics
This is a very long problem of homework.
Definitions:
We start by defining a curve as a continuous function $
\phi :\left[ {a,b} \right] \to \left( {M,d} \right)
$ where M is a metric space with ...
2
votes
2answers
170 views
Describe and illustrate the ball $B_1(0,0) $.
On $\mathbb{R}^2$ we have a metric defined by $d(x,y)=|x_1- y_1|+ |x_2- y_2|$. Describe and illustrate $B_1(0,0)$, the ball of radius $1$ centered at the origin $(0,0)$.
SOLUTION
By definition ...
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3answers
244 views
Example of a homeomorphic map $T:X→Y$
Definition. Let $X$,$Y$ be metric spaces.Then a map $T:X\to Y$ is an homeomorphism if $T$ is continuous, open and bijective.
I don't find a counterexample of such maps, may someone give me at least ...
2
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1answer
160 views
Showing that $d(m,n)=|m^{-1}-n^{-1}|$ is not a complete metric on $\mathbb Z^+$
Let $X$ be a set of all positive integers and define metric $d$ on $X$ by $d(m,n)=|m^{-1} - n^{-1}|$. I'm required to show $(X,d)$ is not a complete space.
SOLUTION:
Let $\{x_n\}$ be any Cauchy ...
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votes
2answers
80 views
To show $X_2$ is complete space
Suppose $X_1$ and $X_2$ are isometric and $X_1$ is a complete space; show that $X_2$ is a complete space. Here I need somebody to help me or to give me ideas.
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1answer
98 views
Complete metric space, with floor function.
I have a problem with this excercise. I need your help.
Let $f:\mathbb{R}\longrightarrow\mathbb{R}$
$f(t)=t+[t]$
where $[\cdot]$ is the floor function.
Define the metric:
$$d(x, ...
