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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50 views

Limit of measurable functions is measurable?

Suppose $(\Omega, \cal F)$ is a measurable space and $(X, \mathcal B_X)$ is a topological space with its Borel sigma algebra. If $f_n: \Omega \to X$ is a sequence of $(\cal F , B$$_X)$-measurable ...
1
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1answer
48 views

Why is $f(x,y)$ said to be discontinuous at $(0,0)$?

Why is $f(x,y)=\begin{cases} \frac{x^2y}{x^4+y^2}, & \text{if $(x,y)\neq (0,0)$}\\[2ex] 0, & \text{if $(x,y)=(0,0)$} \end{cases}$ said to be discontinuous at $(0,0)$? I am supposed to show ...
2
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3answers
32 views

If $K\in \Bbb{R}^n$ is compact, $\sup_{x,y\in K}|x-y|=\max_{x,y\in K}|x-y|$.

Suppose $K\in \Bbb{R}^n$ is compact. Let us denote $D=\sup_{x,y\in K}|x-y|$ as $K's$ diameter. Prove there exist $a,b\in K$ such that $D=|a-b|$ i.e, that the suprimum is the maximum. I know there ...
0
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1answer
36 views

Show that the space $ℓ^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0\text{ for } j>>1\}$ is not complete

Show that the space $$\ell^0=\{\{a_j\}_{j=1}^{\infty}\subset \mathbb C\mid a_j=0 \text{ for } j\gg1\}$$ with inner product $$(a,b) \in ℓ^0\timesℓ^0 \mapsto \langle a,b\rangle =\sum_{j=1}^\infty ...
1
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1answer
24 views

Uniform convergence on an interval.

Let $f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in \mathbb{N}$ Show that there is no uniform convergence on $[1,+\infty[$. I found this particular part of an exercise in my textbook and ...
0
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2answers
43 views

Differentiating $f(x)=\sum_{i=1}^{N}|x-y_i|^2$ where $y_1,…,y_N\in \Bbb{R}^n$.

Let $y_1,...,y_N\in \Bbb{R}^n$ and let $f(x)=\sum_{i=1}^{N}|x-y_i|^2$. I need to show that $f$ has a minimum. I try to differentiate but I am having troubles doing so. First of all, does $|x-y_i|$ ...
0
votes
1answer
24 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
3
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0answers
55 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
2
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0answers
33 views

Does the zeros of an analytic function $f$ form a discrete set? [duplicate]

A function $f$ is called analytic if locally it is given by a convergent power series. Let $ U \subset \mathbb R^n $ be an open set and $f : U \to \mathbb R$ be non zero analytic function. Does the ...
1
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1answer
36 views

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$.

If a set $E\subset \Bbb{R}^n$ is closed and open, then it is either $\Bbb{R}^n$ or $\emptyset$. I have an attempt. I know, or at least think, that it is correct ideally, but I don't know how to make ...
0
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2answers
16 views

Is homeomorphic image of closed bounded subsets of metric spaces , also closed bounded in the homeomorphic image metric space?

Let $X$ , $Y$ be homeomorphic metric spaces with homeomorphism $f$ , then is it true that for any closed bounded subset $A$ of $X$ , $f(A)$ is also closed and bounded in $Y$ ?
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2answers
15 views

Suppose $S$ is a subset of $\mathbb R$ and $x \in \mathbb R$. Show that there are at most two nearest points of $S$ to $x$

Suppose $S$ is a subset of $\mathbb R$ and $x \in \mathbb R$. Show that there are at most two nearest points of $S$ to $x$ Attempt: Suppose there are three nearest points $a,b,c$ of $S$ to $x$. Let ...
0
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2answers
40 views

How can I show uniform convergence?

Let $f_n(x)=\frac{x^n}{1+x^n},~x\geq 0, ~n\in\mathbb{N}$. 1.1.: Determine the pointwise limi of $(f_n)$, $x\geq 0$. 1.2.: Show that the sequence $(f_n)$ is uniformly convergent on ...
0
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1answer
35 views

Uniform continuity.

Check if the mappings $\mathbb{R}\to\mathbb{R},x\mapsto x^2$ and $[0,\infty[:\mathbb{R},x\mapsto \sqrt{x}$ are uniformly continuous. I was going through some old exams our teacher ...
2
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2answers
32 views

Contraction on a metric.

Let $M=\{x\in \mathbb{R}|x\geq 1\}$ with the absolute metric, being a metric space. Show that: a) The mapping $f:M\to M$ with $f:M\to M,~f(x)=\frac{1}{x}+\frac{x}{2}$ is a ...
2
votes
1answer
32 views

A ball with respect to a discrete metric

Suppose we have a ball of radius $1$, then: $$B_{\mathbb{R}^2,d_{disc}}((0,0),1) = \{(0,0)\}$$ But if we increase the radius to be larger than $1$, how is it that the ball would encompass all of ...
1
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0answers
38 views

What logical resources does one need to distinguish $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ from $\mathbb{Q}$?

(I'm inspired by this question.) As a strict lower bound, having just the order relation is not enough, since $\mathbb{R}\hspace{-0.05 in}-\hspace{-0.04 in}\mathbb{Q}$ and $\mathbb{Q}$ are both ...
3
votes
1answer
43 views

No convergence in Discrete metric - why?

If I have a sequence $(x_n)_{n \in \mathbb{N}} := (\cfrac{1}{n},\cfrac{1}{n})$ in $\mathbb{R}^2$ Then why isn't there convergence with respect to the Discrete metric? for a discrete metric, the ...
0
votes
1answer
82 views

Rare metric space proof [on hold]

Let $C = \mathbb R^{\mathbb N}$ be the space of all possible real sequences. Let's define a metric for this space: $$ d((x_1, x_2, \ldots),(y_1, y_2, \ldots)) = \begin{cases} ...
1
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1answer
43 views

Metric Space Definition

From my book, the definition given is: Given a set $X$, a function $d: X \times X \to \mathbb{R}$ is a metric on $X$ if for all $x,y \in X \dots$ Then a metric space is a set $X$ together with a ...
2
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0answers
15 views

Point set where each point has unity distance to all other points ($L_1$ metric)

I want to construct a point set where each point has the same (w.l.o.g., unit) distance to all other points in the $L_1$ metric. Example: The points $\left(\frac{1}{2},0\right)$, ...
3
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0answers
37 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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3answers
33 views

Discrete metric means all sets are countable?

I was working on a proof of "Show that if $A \subseteq \Re^2$ is discrete, then A is a countable set." and I thought about using the discrete metric ($d(x,y)=\delta_{xy}$) on the set as an example ...
0
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0answers
18 views

Computing the Log-Euclidean distance efficiently by using eigen-analysis.

Let $A,B\in\Bbb{S}_{++}^n$ be two symmetric positive definite $n\times n$ matrices with real entries. The Log-Euclidean distance between these matrices is defined as follows $$ d = \lVert \log(A) - ...
0
votes
3answers
79 views

What does it mean for a number to be “larger” than another number? [on hold]

In everyday life we have a clear notion of what it means for something to be larger than something else. Usually we would evaluate something's size based on it's volume. However, is there a formal ...
1
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1answer
38 views

How does this imply “the unit balls completely determine the metric space structure”?

Let $X$ be a metric space. Suppose you know that $$B[x,r] = x + rB[0,1]\qquad\text{and}\qquad B(x,r) = x + rB(0,1)$$ where $B[x,r]$ and $B(x,r)$ are respectively the closed and open balls in $X$. How ...
3
votes
1answer
82 views
+50

Existence of a bounded ball

Lets define: $$F(X) :=\{A \subseteq X \mid A \neq \emptyset , A = \overline{A}\}.$$ For $A, B \in F(X)$ and $p \in X$ define $$d_p(A,B) = \sup_{x \in X} \{ | \operatorname{dist}(x,A) - ...
2
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0answers
9 views

Comparing geodesics on a hypersphere

Background: In Euclidean space, a simple and easy to compute distance metric between two vectors $\mathbf{u}$ and $\mathbf{v}$, is the cosine similarity $ \mathbf{u} \cdot \mathbf{v} / |\mathbf{u}| ...
2
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3answers
37 views

Limit in the set of real sequences.

I have troubles trying to prove the following proposition: Let $S$ be the set of real sequences with ...
-1
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0answers
38 views

A question about discrete topological space [closed]

Let $(X,τ_δ)$ discrete topological space i have a question about it Is it locally compact space or not ?
4
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1answer
41 views

Why are all open subsets not infinite in extent?

I have been looking at the definition of an open set, for a metric space. I have come across the following definition, a few times: An open set $U$ of the metric space $(X,d)$ is a set given that ...
3
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0answers
32 views

Example of a metric group over $\mathbb{R}_0^+$

Do you know an example of a function $d:\mathbb{R}_0^+\times\mathbb{R}_0^+\to \mathbb{R}_0^+$ for which the following properties hold? Or can you prove this does not exist? There exists an $e\in ...
0
votes
2answers
30 views

Cauchy sequences of two metric spaces with only different distance functions

Consider two metric spaces $(X,d)$ and $(X,d_1)$. If given that for a fixed natural number $N$ we have $$\large{d_1(x,y) = \frac{d(x,y)}{N}, \forall x,y \in X}$$, is there any relation between the ...
0
votes
0answers
21 views

Examples of space which is not totally bounded

I know some examples of spaces which are not totally bounded. For example, the real space $R$ with discrete metric is bounded but not totally bounded. I understand its not totally bounded because the ...
0
votes
1answer
65 views

Is this thing a value quantale?

I am currently trying to understand R. C. Flagg's "Quantales and continuity spaces". However I am struggling a bit with his definitions and would like to have a good simple (but not too simple) ...
3
votes
1answer
46 views

Metric on Homogeneous Space $G/H$

For simplicity, assume $G$ is compact and semi-simple Lie group, and $H$ is a closed subgroup of $G$. Therefore the homogeneous space is reductvie, say $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ where ...
1
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1answer
36 views

Prove completeness of a metric space

Let $\mathcal{K} = \{A \subset \mathbb{R}^N| A \neq \emptyset, A \text{ closed and bounded with respect to the euclidean metric} \}$ Let us define $A_\epsilon = \bigcup_{x \in A}U_\epsilon(x)$, where ...
4
votes
0answers
41 views

Show that $X-C$ is connected [duplicate]

Let $X$ be a connected metric space, let $A \subseteq X$ be a connected set. Let $\mathcal{C}$ be a connected component of $X-A$. Show that $X-\mathcal{C}$ is connected. Ok, so Im been dealing ...
0
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1answer
61 views

How can I prove that a set is connected?

For example,if $A \subseteq \mathbb{R}^2$ is finite, so $\mathbb{R}^2 \backslash A$ is connected. I'm trying to use the negation of the definition of connected metric space , so can I reach a ...
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0answers
27 views

Construct a SPD kernel using a (true) distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$ be a non-empty set of pairs $(\mathbf{x},\Sigma_x)$, where $\mathbf{x}\in\Bbb{R}^n$, $\Sigma_x\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes ...
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1answer
20 views

Showing that compact metrizable abelian group has enough characters

I wanted to know if anybody has some clue as why a compact metrizable abelian group has enough characters to sepatare points. (a character on a topological group is a continuous homomorphism from the ...
1
vote
1answer
20 views

Metric spaces - continuity - open/closed.

Let $f:(M_1,d_1)\to (M_2,d_2)$ be a mapping between two metric spaces. a)Let $A\subseteq M_1$ be open and $B\subseteq M_1$ closed. Show through the use of counterexamples that in general ...
1
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0answers
21 views

countable dense set of space of continuous functions on a campact set

Let $X$ be a compact metric space. Let $C_+(X)$ be the set of all non negative continuous functions on $X$. Do there exist a countable dense set of $C_+(X)$? I think the answer is affirmative. For ...
2
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1answer
37 views

Metric spaces as Cauchy complete categories, nlab entry, insight into a few of the constructions.

I'm having a bit of trouble making sense of some of the concepts in the "Metric space" section on nlab's entry on "Cauchy complete category" ...
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0answers
35 views

Existence of an homeomorphism between $X$ a complete separable metric space and a subspace of $[0,1]^{\mathbb{N}}$

Result: If $X$ is a complete separable metric space then there is a $E \subset [0,1]^{\mathbb{N}}$ such that $X$ is homeomorphic to $E$ ($E$ is a $G_\delta$ set - is the intersection of denumerable ...
3
votes
1answer
47 views

Connectedness of the Hausdorff distance.

Does anyone know a proof of connectedness of the Hausdorff distance? I mean a proof of the following: Theorem If $(X, \rho )$ is a connected metric space, then $(F(X), d_h )$ is also connected. ...
3
votes
2answers
32 views

What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
4
votes
2answers
27 views

sequence of open sets

Find the sequence of open sets in $\Bbb{R}$ like $\{G_n\}$ such that $\Bbb{Z}=\bigcap _{n=1} ^{\infty}G_n$. I think an answer is this: $$G_n=\bigcup_{m=1} ...
4
votes
1answer
32 views

Does this implies that two metric spaces are Equivalent?

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).Does this imply that the ...
1
vote
1answer
33 views

Proof that $\mathbb Q_p$ is unique up to unique isomorphism preserving the absolute values

On pages 58-59 of Gouvea's $p-$adic Numbers: An Introduction, he gives the following proof that the field $\mathbb Q_p$, constructed using equivalence classes of Cauchy sequences, is unique up to ...