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Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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Sequentially compact metric space is totally bounded.

I want to prove this: " If for any sequence $(x_n)$ from a metric space $(E,d)$ we can extract a convergent subsequence then for any $r>0$, we can cover $E$ by a finite number of open balls of ...
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Why is the p-adic metric an ultrametric?

I am currently working on the proof that the p-adic metric is an ultrametric. I have proven the first two axioms, but I am not quite sure how to prove the "modified" triangle-equation. I have found a ...
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16 views

Convergence of a sequence in a metric space [on hold]

What is a necessary and sufficient condition for a sequence in a metric space to converge?
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21 views

When can an isometry be constructed between to metric spaces?

I wanted to show that the space $\mathcal{S} := \{ x = (x_k)_{k \in \mathbb{N}} \subset \mathbb{C}\}$ with the metric \begin{equation*} d: \mathcal{S} \times \mathcal{S} \to \mathbb{R}, \ (x,y) \...
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2answers
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Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology. [duplicate]

Let $(X,d)$ be a metric space and $ ( x_{n} )$ , $ ( y_{n} )$ convergent sequences in $X$. Is the sequence $( d(x_{n},y_{n}))$ convergent if $X = \mathbb{R}$ with standard topology. I am having ...
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22 views

Show that $\varphi: \overline{E} \times K \longrightarrow N$ is continuous

Let $K$ compact in a metric space $M$, $N$ is a metric space, $\mathcal{C}(K,N)$ the space of continuous functions $f: K \longrightarrow N$ and $E \subset \mathcal{C}(K,N)$ a family of functions such ...
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22 views

Homeomorphism between 2 metric spaces [duplicate]

Is the plane minus four points on the x-axis homeomorphic to the plane minus four points in an arbitrary configuration? In general, how can I show that it doesn't matter which finite points removed ...
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31 views

Understanding the proof of Kechris' theorem 3.11

In Alexander Kechris' Classical Descriptive Set Theory, he proves a quite useful theorem (3.11) that I'm using as a vital part of a project I'm doing. However, there's a part of the proof I can't wrap ...
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24 views

On the invariance of ball $B(a,r)$.

Let $X$ be a complete metric space and a continuous map $ f: X \to X $. A condition sufficient for a ball $ B(a, r) \subseteq X $ to be invariant by $ f $, that is $ f \left (B (a, r) \right) \...
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1answer
36 views

Property of closed balls and closed sets

Let $(X,d)$ a metric space. Prove that the statements are equivalent : $\textbf{1.}$ For all sequence of closed ball $\{B_n\}$ such that: $B_{n+1} \subseteq B_{n}, \forall n\in \mathbb{N}$ and their ...
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Complete Metric Spaces and discrete spaces

$\textbf{Definition:}$ Let $(M,d)$ a metric space. A point $x \in M$ is a isolated point of $M$ if exists $r>0$ such that : $$ B(a,r)=\{a\} $$ A metric space $M$ is called discrete if every ...
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The relationship among different types of fundamental spaces.

I'm just looking to make sure my understanding of certain fundamental spaces are correct. Denote the set of all vector spaces by $V$, the set of all metric spaces by $M$, the set of all normed spaces ...
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3answers
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Is set of isolated points of S closed?

In metric space, $(X,d)$ I know limit points of some set(S maybe) is closed in X Then is set of isolated points of S is closed? Want to make sure if closure of isolated points of S doesn't ...
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1answer
23 views

Homeomorphism or not?

Problem: Let $f:X \rightarrow Y$ and $g:Y \rightarrow Z$ where $X,Y$ and $Z$ are metric spaces. If $g \circ f$ is homeomorphism and $f$ is surjective prove that $f$ and $g$ are homeomorphism. The ...
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1answer
67 views

Why coarse maps have to be proper?

A map $f: X \to Y$ between metric spaces is said to be coarse, if the following two conditions hold: $f$ is bornologous, i.e. $$\forall_{R>0} \; \exists_{S>0} \; d(x,y) < R \Rightarrow d(f(...
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2answers
29 views

An isometric mapping $f:(X,d) \to (X, d)$ is an open mapping

In general, the result doesn't hold. For example $f: \mathbb{R} \to \mathbb{R}^2$ with $f(x)=(x,0)$ [Euclidian Norm]. Now my attempt to prove the restricted case goes as follows: Let $V \subset X$ ...
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Prove that $\textbf{Int}(Y)$ is the largest open set contained in $Y$

In Green and Gamelins book "An Introduction to Topology" there is an exercise Id like some feedback on. If $Y$ is a subset of $X$, then prove that $\textbf{Int}(Y)$ is the union of all open subsets of ...
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1answer
39 views

Proof that l2 has a countable and dense subset

this is my first question and I hope I don't make any relevant mistakes. For a little bit of context, in my real analysis homework I have the following problem. Show that the subset D $\subset$ $l_2$...
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1answer
22 views

Compact subset of the image of a continuous function

Given complete metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, a continuous function $f:X \to Y$, and a compact subset $K \subset f(X)$, I would like to know if it is possible to claim that there exists a ...
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1answer
30 views

Closedness of the Set of Continuous and Increasing functions

Let $ A=\left\{ f\in C\left[ 0,1\right] |\text{ }f\text{ is strictly increasing and }f\left( 0\right) =0\text{ and }f\left( 1\right) =1\right\} $, where $ C\left[ 0,1\right]$ is the set of continuous ...
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31 views

Compactness of a subset of $L_1([0,1])$

Suppose that $X = L_1([0,1]):= \{f:[0,1]\mapsto [0,\infty): \int_0^1 |f(x)|~dx <\infty\}$. Equip $X$ with the standard metric $d(f,g) = \int_0^1 |f(x)-g(x)|~dx$. Now, define: $$C:= \{f\in X: f(x) \...
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1answer
58 views

A metric on non negative symmetric matrix

Let $w=\{w(i,j)\}_{1\le i,j\le m}$ be an $m\times m$ symmetric matrix with non-negative real entries such that $w(i,j)=0$ iff $i=j$. Show that $$d(i,j)=\min\left\{\sum_{j=0}^{k-1}w(i_j,i_{j+1}):k\ge1,...
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Doubt about equivalent metric spaces

In a metric space (X,d) for any x, y ∈ X, d1(x, y) = d(x, y)/(1 + d(x, y)) and d2(x, y) = min{1, d(x, y)}. Show that they are equivalent. I tried to start in the way that Let U is a set open in d1,...
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Count metric induced topologies on countable set. [duplicate]

How many topologically distinct metrizations of countable set are there? One can find countably many different ones by arranging points on a plane for example. Are there more than that? How would one ...
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26 views

Show that any subspace of a compact space can be covered with one open subspace.

Here's the problem I'm dealing with: Let $(X,d)$ be a compact metric space and let $(U_{\lambda})_{\lambda \in \Lambda}$ be an open cover of $X$. Show that there exist $\delta >0$ such that for ...
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1answer
18 views

Verification of proof that $Y$ nowhere dense iff $X\setminus \overline{Y}$ is dense.

Claim: $Y$ is nowhere dense $\iff$ $X\setminus \overline{Y}$ is dense. Proof $Y$ is nowhere dense $\iff$ $Int(\overline{Y})=\emptyset$ $\iff$ $\forall y\in\overline{Y}, \forall r>0, \quad B_r(y)...
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2answers
171 views

Prove every subset of in the discrete metric is clopen

Hey fellow math enthusiasts! I am reading in ”Introduction to Topology” by Gameline and Greene and I got stuck on an exercise in the first chapter, and I’d love some help on understanding their ...
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53 views

Prove that the series $\sum_{n=0}^{\infty}$ $\frac{x^{2n+1}}{2n+1}$ $-$ $\frac{x^{n+1}}{2n+2}$ converges pointwise but not uniformly on $[0,1]$

I am trying to show that $\sum_{n=0}^{\infty}$ $\frac{x^{2n+1}}{2n+1}$ $-$ $\frac{x^{n+1}}{2n+2}$ converges pointwise but not uniformly on $[0,1]$. My current technique is the split the domain into ...
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If $f: X \to Y$ is a quasi-isometry between geodesic spaces, and $\gamma$ is a geodesic segment in $X$, is $f\circ \gamma$ a geodesic segment in $Y$?

If $f: X \to Y$ is a quasi-isometry between geodesic spaces, and $\gamma: [a,b] \to X$ is a geodesic segment in $X$, is $f\circ \gamma$ a geodesic segment in $Y$? I know that $f \circ \gamma$ is a ...
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32 views

Distance between two measures

Suppose we have (practically infinite number of) i.i.d. samples from two probability measures on $\mathbb{R}^d$, not necessarily having a density with respect to the Lebesgue measure on $\mathbb{R}^d$....
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1answer
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Query regarding a compact set in a open set $\Omega$ in $\Bbb{C}$

Suppose $\Omega$ is a bounded open set in $\Bbb{C}$ For any $\delta>0$ let us define a new set $\Omega_\delta=\{z\in\Omega|\overline{D_\delta(z)}\subset\Omega\}$. Is $\Omega_\delta$ compact? ...
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1answer
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Compact Hausdorff space is metrizable if there countable separating continuous functions

Proposition: Let $X$ be a compact Hausdorff space. Suppose there are countable real valued continuous functions $\{f_n\}_{n \in \mathbb{Z}_+}$ separating $X$ i.e. for all $x, y \in X$ with $x \neq y$, ...
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1answer
64 views

Conjecture: Is it true that the limiting diameter of a (strictly !) nested family of compact sets must be 0 ??

Consider a family of compact subsets of $\mathbb{R}^n, C_1 \supset C_2 \supset C_3 \ldots$. Also, and this is the important bit, 1) $C_j$ has empty interior for all $j \in \mathbb{N}$ 2) The ...
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1answer
40 views

Theorem 1.27 in Rudin's Functional Analysis

1.27 Theorem: Suppose $Y$ is a subspace of a topological vector space $X$, and $Y$ is an $F$-space (in the topology inherited from $X$). Then $Y$ is a closed subspace of $X$. Here is Rudin's proof: ...
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Linearization of the Pythagorean theorem with matrices

Suppose the differential relation $ds=\alpha dx+\beta dy$. Squaring each side of the relation, we obtain: $$ (ds)^2=\alpha^2(dx)^2+\beta^2(dy)^2+\alpha\beta dxdy+\beta\alpha dydx $$ The structure of ...
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1answer
19 views

distance between any two points in a metric space and its diameter

Could anyone tell me how to prove the following? $d(x,y)\le \text{diam }(S)^{1-\alpha}\cdot d(x,y)^{\alpha}$ where $S$ is any complete, separable metric space. Or compact metric space. $0<\alpha\...
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0answers
27 views

are these notations same?

$(X,d)$ any metric space, are these notations has the same meaning? $1.$ $d^q(x,y)$ for some $a>0$, by which I mean, first, they are calculating the distance between $x$ and $y$ which a positive ...
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1answer
44 views

Prove that if metric spaces $(X, d_{X})$ and $(Y,d_{Y})$ are complete so is metric space $X\times Y$ complete

a) Metric spaces $(X, d_{X})$ and $(Y,d_{Y})$ are complete. Prove that metric space $X\times Y$ is complete if the metric is defined as: $\rho((x_1,y_1),(x_2,y_2))$ = $\sqrt{d_X(x_1,x_2)^2 + d_Y(...
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3answers
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How $A=\{x_n:n\in \mathbb N\}$ is an infinite set?Can you explain the proof?

Can you please explain the underlined arguments? How does it deduce that $A$ is infinite? I understood that when $y\in X$. There is an $U\in \mathscr O$ such that $x\in U$(since $\mathscr O$ is an ...
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0answers
10 views

two limit points in a T1 quasi metric space

If I know that $$ \lim_{n\to\infty} d(a,x_n)=\lim_{n\to\infty} d(b,x_n)=0 $$ in a $T_1$ quasi metric space, where $\{x_n\}$ is a right Cauchy sequence (meaning that for $n>m$, $d(x_n,x_m)$ goes ...
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2answers
48 views

Can a connected compact subset in $\mathbb{R} ^3$ always be enclosed in a closed ball of minimum radius?

Given any connected compact set $A \subset \mathbb{R}^3$, does there exist a closed ball $B_r (x)$ of radius $r$ around $x\in A$ such that $r$ is minimum and $A\subset B_r (x)$ under usual topology ? ...
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2answers
44 views

Question about details in the Baire Category Theorem.

Here is the version of the theorem being used in my class. It is from the textbook Introduction to Topology by Gamelin and Greene. Beginning of proof Let $\{U_n \}_{n=1}^{\infty}$ be a sequence of ...
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2answers
45 views

Baire's Theorem: Examples for open dense subsets

Theorem (Baire): Let $(X,d)$ be a complete metric space and $(D_n)_{n \in \mathbb{N}}$ a family of open dense subsets of $X$. Then $\bigcap_{n \in \mathbb{N}} D_n$ is also dense in $X$. This is the ...
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2answers
31 views

Prove that the set $ D=\{(x,y): 0<x^2+y^2<4\} $ is open. Apply the definition.

I tried this: Denoting $P=(x,y)$, we see that $0<x^2+y^2=||P||^2<2^2 \Rightarrow ||P||<2$ Let $(s,t)=Q \in B_{2-||P||}(P) \Rightarrow ||Q - P|| < 2 - ||P||$ And $||Q|| - ||P|| \leq ||Q ...
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3answers
70 views

I am not sure how to finish the proof of showing that the intersection of finitely many open subsets is open.

I am trying to prove the following: Let $(X,d)$ be a metric space. An intersection of finitely many open subsets of $X$ is open. The following is my attempt: Let $x \in \bigcap_{i=1}^n U_i$ be ...
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1answer
56 views

surface area of a solid of revolution. (request for a source for rigourse proof )

link: Doubt in Application of Integration - Calculation of volumes and surface areas of solids of revolution. I know that this question has been asked and answered before , but non of the answers ...
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1answer
32 views

[Baby Rudin's proof]. Continuity of mapping in metric space

The theorem 4.8. states that : A mapping $f$ of s metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$ The ...
2
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0answers
38 views

Let $G$ be a “Normed” Abelian Group, is this a topological equivalent Norm over $G$?

Let $G$ be an abelian group. We say $\Vert \cdot \Vert : G \rightarrow \Bbb R_{\ge0}$ is a norm if it satisfies $\Vert x \Vert =0 \Leftrightarrow x=0$ $\Vert -x \Vert = \Vert x \Vert$ $\Vert x+y \...
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1answer
30 views

intersection of decreasing compact sets is again compact

I am following a real analysis course and I would like to discuss some definitions and results about metric spaces. First of all, the definition of compact set. I the course we defined it as follows: ...
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2answers
26 views

Convergence of a sequence in $\mathbb{R}^p\times\mathbb{R}^q$ and $\mathbb{R}^{p+q}$

Show that a sequence $(x_1^n,x_2^n)$ in $\mathbb{R}^p\times\mathbb{R}^q$ converges to $(x_1,x_2)$ iff the same thing happens when we consider the sequence as belonging to $\mathbb{R}^{p+q}$. I am ...