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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Continuity of the metric function
Let $(X,d)$ be a metric space. How to prove that for any closed $A$ a function $d(x,A)$ is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof:
$$
|d(x,a) - ...
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Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
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If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
Let $A, B$ be subsets of a metric space $X$. If $A$ is compact and $B$ is closed, show that the distance between $A$ and $B$ is achieved.
Attempt at a proof:
Let $A$ be compact and $B$ be ...
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4answers
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$\pi$ in arbitrary metric spaces
Whoever finds a norm for which $\pi=42$ is crowned nerd of the day!
Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way?
For ...
6
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4answers
884 views
Equivalent metrics determine the same topology
Suppose that there are given two distance functions $d(x,y)$ and $d_1 (x,y)$ on the same space $S$. They are said to be equivalent if they determine the same open sets.
Show that $d$ and $d_1$ are ...
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A and B disjoint, A compact, and B closed implies there is positive distance between both sets
Claim: Let $X$ be a metric space. If $A,B\in X$ are disjoint, if A is compact, and if B is closed, then $\exists \delta>0: |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$.
Proof. ...
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What operations is a metric closed under?
Suppose $X$ is a set with a metric $d: X \times X \rightarrow \mathbb{R}$. What "operations" on $d$ will yield a metric in return?
By this I mean a wide variety of things. For example, what functions ...
6
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1answer
342 views
Sum of Cauchy Sequences Cauchy?
Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
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3answers
623 views
Notions of equivalent metrics
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...
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219 views
Show that $d$ is a metric on $\mathbb C^n$
On $\mathbb C^n$, define $||z||=(\sum_{j=1}^{n}|z_j|^2)^\frac{1}{2}$ and for $x,z\in\mathbb C^n$ define $d(z,w)=||z-w||.$ Prove that $d$ is a metric on $\mathbb C^n$.
My attempt:
I need to show ...
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Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
7
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1answer
171 views
If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected
I have an exercise found on a list but I didn't know how to proceed. Please, any tips?
Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of ...
7
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2answers
373 views
Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
I have to show that the real numbers equipped with the metric
$ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space.
Certainly, I have to search for a cauchy sequence of real numbers ...
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1answer
520 views
Prove the map has a fixed point
Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$,$\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$.Prove A have a unique fixed point in K.
The uniqueness is easy.My ...
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Proof of the Lebesgue number lemma
I want to prove the Lebesgue number lemma:
Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, there exists $\delta \gt 0$ such that for each subset of $X$ ...
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1answer
174 views
which of the following metric spaces are complete?
Which of the following metric spaces are complete?
$X_1=(0,1), d(x,y)=|\tan x-\tan y|$
$X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$
$X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$
$X_4=\mathbb{R}, ...
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votes
4answers
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An open ball is an open set
Prove that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open.
My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an ...
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1answer
395 views
Why are metric spaces non-empty?
I'm just second-marking some exam scripts, and I wanted to leap on a question and made the following pedantic remark concerning the model answers: "if the metric space is empty then this proof doesn't ...
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2answers
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Is Completeness intrinsic to a space?
Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric?
If ...
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2answers
202 views
Continuous functions between metric spaces are equal if they are equal on a dense subset
If two functions defined on metric spaces $X$ and $Y$ are equal on a dense subset of $X$ and are continuous also, then are they equal on all of the metric space $X$?
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1answer
74 views
Let $L_p$ be the complete, separable space with $p>0$.
Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
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1answer
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Question(s) about uniform spaces.
I was reviewing questions and notes related to uniform spaces and came across this interesting statement: Every metric space is homeomorphic, as a topological space, to a complete uniform space.
It ...
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630 views
Explanations of Lebesgue number lemma
From Planetmath:
Lebesgue number lemma: For every open cover $\mathcal{U}$ of a compact metric space $X$, there exists a real number $\delta > 0$ such
that every open ball in $X$ of radius ...
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2answers
744 views
if every continuous function attains its maximum then the (metric) space is compact
Suppose $(M,d)$ a metric space. I want to show that if every continuous real-valued function on $M$ attains a maximum, then the space must be compact.
I was trying to do this by assuming $M$ ...
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1answer
795 views
Understanding the idea of a Limit Point (Topology)
I have attached an image of how I was visualizing a limit point, but I'm now not so sure that I have understood the concept correctly after attempting to really draw out what I was visualizing.
I'll ...
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Examples of non symmetric distances
It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of ...
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votes
3answers
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Examples of function sequences in C[0,1] that are Cauchy but not convergent
To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
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1answer
350 views
The space of continuous, bounded functions from a metric space $X$ to $\mathbb R$
Let $(X,d)$ be a metric space. We denote by $C_b(X;\mathbb{R})$ the space of continuous and
bounded functions from $X$ into $\mathbb{R}$, equipped with the sup-norm metric. We define a
mapping $O: X ...
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Continuous extension of a uniformly continuous function from a dense subset.
I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous ...
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2answers
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What are some motivating examples of exotic metrizable spaces
Among topological spaces, the metric spaces are usually considered to be the tame animals. Describing the topological notion of closeness by a distance is so intuitive (as opposed to the abstract ...
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1answer
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Extending a homeomorphism of a subset of a space to a $G_\delta$ set
I am having trouble figuring out the following question (3.10 in Kechris, Classical Descriptive Set Theory): If $X$ is completely metrizable, and $A\subseteq X$ with $f:A\to A$ a homeomorphism, then ...
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The Class of Non-empty Compact Subsets of a Compact Metric Space is Compact
This is a question from my homework for a real analysis course. Please hint only.
Let $M$ be a compact metric space. Let $\mathbb{K}$ be the class of non-empty compact subsets of $M$. The ...
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votes
2answers
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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 ...
5
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2answers
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Do results from any $L^p$ space for functions hold in the equivalent $\ell^p$ spaces for infinite sequences?
For e.g., is $\ell^2$ self-dual like $L^2$? If some $x[n]\in\ell^1\cap\ell^2$, then does it have a Fourier transform in $\ell^2$?
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1answer
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Completion of Topological Group with Metric
Related to this question, I'm having trouble understanding the construction of the completion of a topological group with metric structure. In particular, under what conditions is the completion also ...
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1answer
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Cantor set is boundary of regular open set
Does there exist a regular open set $U$ in $[0,1]$, such that Cantor set is the boundary of $U$?
3
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1answer
149 views
how to show that $c_0$ is complete
I want to show that the metric space $(c_0,d_\infty)$ is complete, where $c_0$ is the collection of all sequences $x\colon \mathbb N\to\mathbb R$ which tend to $0$. I have already shown that the space ...
3
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1answer
267 views
Open and closed balls in $C[a,b]$
Let $X$ be a non empty set and let $C[a,b]$ denote the set of all real or complex valued continuous functions on $X$ with a metric induced by the supremum norm.
How to find open and closed balls in ...
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1answer
175 views
Long proof of equivalence of subspace and metric topology
Let $(X,d)$ be a metric space and $S\subseteq X$. Let $\tau$ be the topology on $X$ induced by $d$ and $\tau_S$ be the subspace topology on $S$:
$$
\tau_S = \{S\cap V:V\in \tau\}.
$$
Denote $B_r(x)= ...
2
votes
2answers
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Metric Spaces Analysis
Let $(X,d)$ be a metric space and for $x,y \in X$ define
$d_b(x,y) =$ $ \dfrac{d(x,y)}{1 + d(x,y)}$
a) show that $d_b$ is a metric on $X$
Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$
...
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3answers
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Metric space where the distance between arbitrary points is a constant
Can I have a metric space where the distance between two points is an arbitrary constant? Does this mean that there cannot be 'co-linear' points in the space? i.e. if A B and C are colinear, and B is ...
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2answers
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A subset $G$ of $\mathbb{R}^n$ is open iff the complement of $G$ is closed
So I'm covering material for my upcoming final exam, and I have a sneaking suspicion that my teacher will ask us to prove the following theorem:
A subset $G$ of $\mathbb{R}^n$ is open iff the ...
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1answer
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A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded
The title is a claim my classmate made during our summer vacation :D
He showed me a TeX file describing a proof of his claim, and it contains a fairly short but elegant proof. He says that the ...
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1answer
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Continuous function on a compact metric space is uniformly continuous
I am struggling with this question:
Prove or give a counterexample: If $f$ is a continuous function on a compact
subset $Y$ of a metric space $X$, then $f$ is uniformly continuous on $Y$.
...
6
votes
1answer
317 views
Urysohn's function on a metric space
Let $(X,d)$ be a metric space and $A\subset B\subset X$. $A$ is closed, $B$ is open. If there are developed methods to find at least one (or describe the whole class) of Urysohn's functions for $A$ ...
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2answers
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Is a finite union of bounded sets bounded in any metrical space?
In any metrical space $(M,d_M)$, consider $n$ bounded subsets $S_i\subset M$. Then, is $\cup_i^nS_i$ bounded? If so, why?
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1answer
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Metrizable compactifications
Suppose $X$ is a metric space. When does it have a metrizable compactification?
Of course it is enough to discuss complete metric spaces, but separability may not be assumed here.
I know that ...
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2answers
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Are these subsets of $\mathbb{R}$ homeomorphic?
Consider the following subspaces of $\mathbb{R}$ with the usual topology:
$$X = (0, 1) \cup \{2\} \cup (3, 4) \cup \{5\} \cup \cdots \cup (3n, 3n + 1) \cup \{3n + 2\} \cup\cdots$$
$$Y = (0, 1] \cup ...
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2answers
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For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous? [duplicate]
Possible Duplicate:
Continuity of the metric function
For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous?
I think it is, but I'm having ...
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3answers
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the pseudometric induced by a measure
Let $(X, \Sigma, \mu)$ be a measure space.
We can define a pseudometric $d$ on $\Sigma$ in the following way:
$$d(A, B) = \mu(A\bigtriangleup{}B)$$
where $A\bigtriangleup{}B = ...
