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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Lipschitz does not imply fixed points

I have the following problem in mind: Let us say we have a function $f:X\rightarrow X$ (X is a complete metric space) and it respects that if $x\neq y$ then : $d(f(x),f(y))<d(x,y)$. My trouble ...
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1answer
30 views

Are these metrics complete?

Determine if these subsets of R are complete with the Euclidean metric? a) $[0,\infty)$ b) $(0,\infty)$ I know the definitions of completeness and I know the Euclidean metric, but don't know how to ...
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1answer
31 views

Prove that if $B(x,r)$ and $B(x',r')$ are disjoint $\Longleftrightarrow d(x,x') \ge r+r'$

Assuming that $d(x,x') \ge r+r'$, and proving that they are disjoint is easy. It's the other side that I'm having difficulty with. This seems like a really easy problem, but i'm having difficulty ...
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3answers
43 views

Closed and Connected Subset of a Metric Space

My English may not be perfect since I'm not a native speaker, so please do point out the grammar mistakes if there are any. I've been reading Conway's "Functions of One Complex Variable", and ...
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3answers
47 views

Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
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4answers
45 views

(Proof Checking) Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$.

Prove that if $C,C'$ are compact subsets of a hausdorff space $X$ then $C\cap C'$ is compact in $X$. I am tempted to use the following argument. Let $U = \{U_i|i\in I\}$ be some open cover of $C\cap ...
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1answer
24 views

find a sequence of closed connected subsets $V_n$ of $\mathbb R^2$ s.t. $V_n\supseteq V_{n+1}$ and $\cap^{\infty}_{i=1}V_i$ is not connected

Find an example of a sequence $V_n$ of closed and connected subsets of the Euclidean plane satisfying $V_n \supseteq V_{n+1}$ such that $\cap_{i=1}^{\infty}V_i$ is not connected. Normally I would ...
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1answer
18 views

Hausdorff distance for empty sets?

The Hausdorff distance is defined for non-empty sets. What would be a reasonable generalization of the definition for the case when one of the sets is empty, if the generalized distance should remain ...
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29 views

Closure of a particular set, $c_{00}$.

I have been learning the concept of closure in metric spaces and all has made sense so far. However, I have come across a particular example that is troubling me in terms of extracting the possible ...
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1answer
11 views

Closure of interior proper subset of interior of closure

$\newcommand{\intr}{\mathrm{int}}$ I need to give an example of a metric space $(X,d)$ and $A ⊆ X$ so that $\overline{\intr(A)} ⊂ \intr (\overline{A})$, where $\overline{B}$ refers to the closure of ...
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32 views

How much close should two spaces be in the Gromov-Hausdorff distance to be homeomorphic?

Here I am considering the Gromov-Hausdorff convergence for metric spaces. I know two compact metric spaces are isometric if and only if $d_{GH}(X,Y)=0$, where $d_{GH}$ denotes the Gromov-Hausdorff ...
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0answers
61 views

Prove that a map is a homeomorphism and the inverse is bounded

I'm trying to unravel an obscure passage in a textbook, which states that if $\phi :\mathbb{R}^m\to\mathbb{R}^m$ is continuous, bounded and Lipschitz with constant $\varepsilon$ (which is still free ...
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1answer
28 views

proving the equivalence of to metrics

Any hints as to how I can prove that $(\mathbb{R}^n,d_\infty)$ and $(\mathbb{R}^n,d_T)$ are topologically equivalent. Where $$d_\infty = \sup\{|x_i-y_i|, |x_2-y_2|,..,|x_i-y_i|\} $$ and $$d_T = ...
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1answer
41 views

Proving a basis exists

How would I show that there exists some set of open balls with rational radius and rational centre such that they are a subset of the reals.That is, $\exists p,q\in \mathbb{Q} $ and $ r,x \in ...
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4answers
893 views

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Example of two open balls such that the one with the smaller radius contains the one with the larger radius. I cannot find a metric space in which this is true. Looking for hints in the right ...
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1answer
46 views

The “intersection property” of the symmetric difference metric

$\newcommand{\measure}{\operatorname{measure}}$ The symmetric difference between sets can be used to define a pseudo-metric on the set of subsets of a given measure space: $$d(S,T)=\measure(S\oplus ...
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1answer
21 views

Verify why this is not a metric $d(x,y)=\|x-y\|_p$ ( $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$).

$d(x,y)=\|x-y\|_p$ $p$ is prime and $\|x\|_p=p^{-h}$ if $x=p^h\dfrac{m}{n}$, where $m, n$ are coprimes with $p$. This is not a metric because if $x=y=p^k\dfrac{m}{n}$, then $x-y=0=p^0\dfrac0n$. ...
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3answers
53 views

proving topological equivalence

How would I show that $$d_E = \sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$ and $$d_\infty= \sum_{i=1}^n|x_i-y_i|$$ and $$\sup\{|x_1-y_1|,|x_2-y_2|,...,|x_n-y_n|\}$$ are topologically equivalent on $\mathbb{R}^n$? ...
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2answers
76 views

A metric space is compact if and only if its complete and totally bounded.

I just need help with one direction. In particular suppose that $(X,d)$ is complete and totally bounded. Suppose that $X$ is infinite. Take any infinite subset $Y$ in $X$ and then just show that $Y$ ...
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2answers
46 views

proving a metric

I'm trying to show that given a metric $d(x,y)$ show that $d_0(x,y) =\frac{d(x,y)}{1+d(x,y)}$ is also a metric.. It's trivial to show the first two properties, that is, $d_0\geq 0 $ & for $x=y ...
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1answer
64 views

Hausdorff distance and intersection

The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set. Suppose there ...
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0answers
23 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
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1answer
26 views

Convergence of circles in Hausdorff distance

Every triple of real numbers $(x,y,r)$, where $r>0$, represents a circle with center $(x,y)$ and radius $r$. Suppose we have a infinite sequence of such triples $(x_i,y_i,r_i)$, that converges to ...
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0answers
28 views

metric makes $\mathcal{C}(\Omega) $ to a complete metric space

I have some problem with the second part Show that $$\rho(f,g):= \sum_{m=1}^\infty \frac{1}{2^m} \frac{\max_{x \in \bar{\Omega}_m}|f(x)-g(x)|}{1+\max_{x \in \bar{\Omega}_m}|f(x)-g(x)|}$$ is a metric ...
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1answer
27 views

Endowing a metric on the torus from the euclidian metric of its covering space, the plane

In Thurston's and Levy's "Three dimensional Geometry and Topology, page 6, they define the induced metric on the torus from the euclidian metric of its covering space, the plane. Specifically, for ...
3
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35 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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1answer
31 views

Borel-set, open, measurable function.

I have a questions about Borel sets. Here is how they defined in my book: Now they say that, the set consits of open sets. But it must not nececarrily be all open sets on X? The reason this ...
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19 views

Hairy ball theorem, projections and L.I. vectors

I'm trying to understand this paper which proves that not every unimodular row is completable by invertible matrices: Why we have these implications: There are two linearly independent vectors at ...
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22 views

Prove the $d_\infty$ metric is finite

I need to show that the $d_\infty$ metric $$d_\infty(x,y) = \sup|x_i-y_i|$$ for all sequences $x$ such that $\sup|x_i| < \infty$ is indeed a metric by checking the metric axioms. I also need to ...
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23 views

Prove (X, l2) is indeed a metric space

I am having some trouble proving that little l2 is indeed a metric. I am getting stuck at proving the triangle inequality holds. d2(x,y)= (∑|xi-yi|^2)^0.5 So I started off by "adding zero" |xi-zi| = ...
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1answer
33 views

If $X$ is complete, then every subset of $X$ that is closed and totally bounded is also compact. Does the converse hold?

Let $X$ denote a metric space. Supposing every subset of $X$ that is closed and totally bounded is also compact, is $X$ necessarily complete? What I've got so far. Assume every $X$ subset of $X$ ...
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1answer
28 views

Suppose two metrics induce the same bornology; do they necessarily induce the same topological space?

Every metric space $(X,d)$ gives rise to a bornology by asserting that $A \subseteq X$ is bounded iff there is an open ball (of finite radius) that includes $A$. Now just because two metrics induce ...
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2answers
46 views

Proving there is a sequence convergent to a limit point of a set without axiom of countable choice?

Often, we use a construction like this: Given a subset $ A $ of a metric space and its limit point $ a $, we know that for every $ \epsilon > 0 $ there is another point $ x $ different from $ a $ ...
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24 views

Differentiation of Radon measures

Assume $\ (X,d)$ is a locally compact metric metric space and $\ \nu,\, \mu$ are Radon measures on $X$. Then, suppose that the following hypothesis hold: $\ w\in ...
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Question about finite subcovers

I'm having problems wrapping my head around the part with $\rho_i$.Here goes: $A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely ...
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1answer
24 views

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and vice versa

Give an example of a metric space $(X,d)$ and $A\subseteq X$ such that $\text{int}(\overline{A})\not\subseteq\overline{\text{int}(A)}$ and ...
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32 views

a complete space

Define a set $$X=\left\{f:\mathbb{R}\rightarrow\mathbb{R}|f \mbox{ is n-times continuously differentiable}\right\}$$ equipped with the norm $$||y||=\max_{\begin{subarray}{l} ...
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0answers
25 views

Some fundamental relations in topology

Are the following relations correct? $\ \{ Normed\, Vector\, Spaces\} \subset \{Topological\, Vector\, Spaces\} \subset \{Uniform \,Spaces\} \subset \{Topological\, Spaces\}$ Then $\ \{Normed\, ...
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1answer
68 views

Metrics and Continuous Functions

The following question reads as i) Given an example of infinite metric spaces $(X,d)$ and $(Y,\delta)$ such that every function $f\colon X\to Y$ is continuous. ii) Is it possible to give an ...
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1answer
47 views

Are there path-connected but not polygonal-connected sets?

This question came up in my mind. In the scope of normed spaces, does there exist a path-connected but not polygonal-connected set? I'd rather say no for open sets (my intuition is that ...
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0answers
36 views

NonCountable Product of Discrete Space {0,1}

Is the product of $\{0,1\}^I$ for any $I$ metrizable (with the product topology)? It Would be helpful to see proof\disproof idea. Thank you
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Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
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1answer
23 views

Compact homeomorphic non bilipschitz homeomorphic metric spaces

Statement: Let $(X,d)$, $(Y,r)$ be homeomorphic compact metric spaces. Then, there exists a bilipschitz homeomorphism between the two. Problem: As I have no clue whether the statement is true or ...
3
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1answer
55 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
3
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1answer
33 views

Pseudometrics and non-expansive maps

Let $X$ be any set, and $(Y,\rho)$ be a pseudometric space. For a collection of functions $F\subseteq Y^X$ define $$ d_{F,\rho}(x',x''):=\sup_{f\in F}\rho(f(x'),f(x'')). $$ It is easy to show that ...
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1answer
17 views

is a space with discrete metric space complet or not?

I have a question about discrete metric spaces: prove that Every discrete metric space $X$ with discrete metric, ($X$,$d_0$) is complete?
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1answer
132 views

Uniqueness of curve of minimal length in a closed $X\subset \mathbb R^2$

Suppose $X$ is a simply connected closed subset of $\mathbb R^2$. Let $a,b$ belong to $X$. Is it true that there is at most one curve inside $X$ from $a$ to $b$ such that the length of the curve is ...
6
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63 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
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3answers
30 views

Limit in metric space

Suppose $x_n \to x$ and $y_n \to y$ in the metric space $\big( M,p \big)$. Prove that $lim_{n \to \infty} p(x_n, y_n) = p(x,y)$ Well, I am not sure how to get this from the properties of the metric ...
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50 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...