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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Is this set open in the Euclidean topology on the plane?

Let $\mathbf{R}^2$ be the two-dimensional Euclidean space, and let $$ A := \{ (x,y) \in \mathbf{R}^2 | \, \, \, |x| < \frac{1}{y^2+1} \}.$$ Then how can we establish (preferably using the ...
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In a metric space are the terms $bounded$ and $totally$ $bounded$ interchangeable?

The question is: Is there a theorem that says in a metric space a set is bounded if and only if it is totally bounded?
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Metric on Riemannian manifolds

Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum ...
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whether they are complete metric space

$S=\{x\in\mathbb{R}: 2x^2\cos{1\over x}=1\}$ $T=\{x\in\mathbb{R}:2x^2\cos{1\over x}\le 1\}\cup\{0\}$ I need to tell whether they are complete metric space under usual metric. As they are closed ...
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Open set in a metric space is union of closed sets

Show that every open set A is in a metric space (X,d) is the union of closed sets. This is a question on my analysis homework. I understand that this can only be true if we consider the union of ...
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Separability almost everywhere

Let $(X,d)$ be a metric space and $\mu$ a Borel probability measure. Suppose that for every $\epsilon>0$ we have that $\mu(B_{\epsilon }(x))=c_{\epsilon}>0$ a.e. Is this enough to show that ...
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2k views

Real Analysis: Prove a set is closed

Let R be equipped with the usual Euclidean metric. Show that the set $S=\{(x,y) \mid 0 \leq y\leq x.\}$ is closed. The approach i am trying is to prove the complement is open, which would imply ...
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Non-completeness of $p$-adic metric on $\mathbb{Z}$

On $\mathbb{Z}$, we define the $p$-adic metric $d_p$ (for $p$ prime) as follows, for $m,n \in \mathbb Z$: If $m=n$ then $d_p(m,n) =0$ If $m \neq n$ then $d_p(m,n) = \tfrac{1}{r+1}$ where $p^r \mid ...
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How can I prove that the closure of a set in a metric space is the set of all limits of sequences in that set?

Let $(X,d)$ be a metric space and let $A \subsetneqq X$. Let $E$ be the set of all $p$ in $X$ for which there exists a sequence $(p_n)$ in $A$ such that $ p = \lim\limits_{n\to \infty}p_n$. Show that ...
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definition of metric space

from the actual definition of metric space ,we know that metric space - a set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric ...
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265 views

The continuity of a distance function

Let $(X,d)$ be a metric space $A\subset X$ be a nonempty subset. The distance function $f : X \to\mathbb R$ by $f(x)=d(x,A)$ where $d(x,A) = \inf_{a\in A} d(x,a)$ and $\mathbb{R}$ denotes the set of ...
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150 views

When is $(f(x),d)$ a complete metric space?

$(X,d)$ is a complete metric space with a metric $d(x,y)=|x-y|$. $f$ is a continuous function. What condition f should satisfy such that $(f(x),d)$ is a complete metric space? I think $f$ is a ...
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368 views

Proving the every subset of $M$ is clopen.

Let $M$ be a metric space with the discrete metric, or more generally a homeomorph of $M$. How can I prove that every subset of $M$ is clopen?
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176 views

let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove the exist an $r>0$ such that $d(x,K)\leq r \rightarrow x \in U$

I'm stuck... I would appreciate some help let $K \subset U \subset X$, $(X,d)$ metric space $U$ open and $K$ compact, prove there exists an $r>0$ such that $d(x,K) \leq r \rightarrow x \in U$
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Convergence of sequences in a finte product of metric spaces.

I am merely looking for solution verification/feedback on the following proposed solution. Thank you!! :) $\textbf{Problem:}$ Let $\{X_1, \ldots , X_k\}$ be a finite collection of metric spaces. ...
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Property of a metric in the space of all the sequences of real numbers

A few weeks ago I had this problem, the adjoint-teacher solved it on class, and I thought I understood, but now I'm rechecking and there are a few things that aren't clear for me. So we defined this ...
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Can this intuition give a proof that an isometry $f:X \to X$ is surjective for compact metric space $X$?

A prelim problem asked to prove that if $X$ is a compact metric space, and $f:X \to X$ is an isometry (distance-preserving map) then $f$ is surjective. The official proof given used ...
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320 views

Regarding this proof that the union of bounded sets is bounded (generalized metric space)

http://www.proofwiki.org/wiki/Finite_Union_of_Bounded_Subsets I am having trouble seeing how this proves the idea at all. It says: $$d(x, a_1) \le d(x, a_2) + d(a_1, a_2)$$ so $$d(x, a_1) \le K_2 + ...
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99 views

A complete ring with respect to an ideal

I would like to know what does it mean to say " a ring $R$ is complete with respect to some ideal $I \subset R$. Is it like, we define a metric by saying two elements in $R$ is "close" if the ...
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48 views

Show that the product of complex matrices is a continuous mapping

Let $\mathbb{M}(n, \mathbb{C})$ be the space of all complex $n\times n$ matrices with operatornorm. On $\mathbb{M}(n, \mathbb{C}) \times \mathbb{M}(n, \mathbb{C})$ we define the product metric, ...
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536 views

Proving that Union of a finite number of complete subsets of metric space $(X,d)$ is complete.

We have following 3 definitions. Definition: Suppose that $(X,d)$ is a metric space. A sequence $(\textbf{x}_n)_{n\in \mathbb{N}}$ of points in $X$ is said to be a Cauchy sequence, if, given any ...
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Complete but not totally bounded metric space with a certain property

Give an example of a complete metric space $(X,d)$ and a nested sequence of nonempty closed BALLS $A_n = \bar{B}(x_n,r) = \{y \in X : d(x_n,y) \leq r\}$ such that $\bigcap_n A_n = \emptyset$. So ...
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Proving openness of sets(simple question)

I'm having trouble proving that a given subset is open. Let's say I'm asked to prove the following set is open. $$A =\{(x,y) : -1 < x < 1, -1 < y < 1\}$$ let $(x_0,y_0) \in A$, then $ ...
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139 views

In a closed subspace A of a complete metric space, are there two points such that $d(x,y) = \mathop{Diam}(A)$

If we have a complete metric space $X$ and $A$ is closed, bounded subspace, do there necessarily exist points $x,y \in A$ such that $d(x,y) = $ Diam$(A)$? Clearly if $X$ is not complete we cannot ...
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150 views

Are all metric space as a euclidean space?

I believe that all euclidean space is a metric space? But I need to know about inverse? I mean: are all metric space as a euclidean space? Is there any kind of metric space which is not euclidean ...
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58 views

Are $d_{\infty}$ and $d_{p}$ distance functions?

Let $X$ be a set equipped with a metric $d_x$, denoted by $\langle X,d_x\rangle$, and $Y$ equipped with a metric $d_y$, denoted by $\langle X,d_y\rangle$. Let $Z=X\times Y$. Let ...
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62 views

A basic doubt on open set

I see in a book the following is claimed : If $E$ is an uncountable subset of $\Bbb R^n$ and then $S \cap E$ is open where $S$ is open. Why ? The intercept from the book is as follows : $E$ is an ...
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A basic question on denseness in a metric space

Suppose we have a metric space with a countable dense subset. Now consider a subset $E$ and a $p \in E$. Now consider the set $\cup_{q \in E, q \neq p}N_{r_i}(q)$. Does it set covers $p$, $\forall r_i ...
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Which of these topological properties imply which?

I am going through the chapter on compactness and completeness from Sternberg's Advanced Calculus and trying to build an intuition for what many of this topological properties mean, and which imply ...
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76 views

A problem on metric space and countable base

Suppose, in a separable (so it has a countable base) metric space $X$, a subset $E$ is uncountable. Now, $P$ be the set of all points $p$ of $X$ such that every neighbourhood of $p$ contains ...
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68 views

Reflection along subspace

A symmetric space is a Riemannian manifold M with the following property: For every point $p \in M$ there is an isometry $\phi: M \rightarrow M$ such that $\phi(p) = p$ and $\phi_*(v_p) = -v_p \in ...
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Open set of $\Bbb R$ which is not bounded below can be written as atmost countable collection of disjoint segments

Suppose I have a open set of $\Bbb R$ which is not bounded below but bounded above. Now, I want to show that that open set can be written as atmost countable collection of open intervals. I have ...
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Group theory with analysis

I've studied group theory upto isomorphism. Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems. I too have done metric spaces and real analysis properly. ...
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60 views

Is the strong convergence of Borel probability measure metrizable?

In a metric space $(X,e)$, a sequence of Borel probability measure converges strongly, $\mu_i \xrightarrow{s} \mu$, iff for each Borel subset $S \in X$, we have $\lim_{i \to \infty}\mu_i(S) = \mu(S)$. ...
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Proof of the completion of a metric space using cantors diagonal argument and showing a diagonal sequence is cauchy

I am studying applied functional analysis out of Applied Analysis by John Hunter. In chp. 1 of the text it gives a proof for the completion of a metric space. I am having trouble with understanding ...
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A problem on countable dense subset on a metric space

Suppose for a metric space every infinite subset has a limit point. What should be my strategy to construct a countable dense subset there? Additionally, how do I intuitively guess that with such a ...
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1answer
90 views

A question on Cauchy sub-sequences in a metric space $(X,d)$

Let $(X,d)$ be a metric space, and let $(x_n)$ be a sequence in $X$. Prove that if $(x_n)$ has a Cauchy subsequence, then for any decreasing sequence of positive $\epsilon_k \rightarrow 0$, there is a ...
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Prove that $\bar A=\bar B \iff d(x,A)=d(x,B)$ for every $x\in\mathbb{R}^n$.

Prove that $\bar A = \bar B \iff d(x,A)=d(x,B)$ for every $x\in\mathbb{R}^n$. There is a lemma which states that $\exists x_0\in \bar A$ such that $d(x,A)=d(x,x_0)$. So for the forward direction (to ...
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331 views

Show that $\mathtt d(A,B)>0$ if $A\cap B=\varnothing$, $A$ is compact and $B$ is closed.

I have this problem: If $X$ is a metric space, and if $A,B\subseteq X$ then the distance between $A$ to $B$ is $\mathtt d(A,B):=\inf \{d_X(x,y)| x\in A,y\in B\}$. Show that $\mathtt d(A,B)>0$ ...
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If $U\subseteq X$ is open in $X$, then $U\cap Y$ is open in $Y$

If $X$ is a metric space, and $Y$ is a metric subspace of $X$ the show that if $U\subseteq X$ is open in $X$, then $U\cap Y$ is open in $Y$. So we have two cases: if $U\cap Y=\varnothing$ and $U\cap ...
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681 views

Open and closed sets in a metric space

I'm having a little trouble understanding the definition of open and closed set in a metric space. I'm going to use an excersize: Lets consider $Y:=(-1,1]$ with the usual metric of $\Bbb R$. ...
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29 views

Algorithm to compute differences in daytime using Euclidean distance

This question came into my mind, when reading this post on how to cluster data with respect to what time of the day it was observed. Say you have a number $h$, where $h\in\{0,\cdots,23\}$. Is there ...
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Relation between dense subset and cover of a metric space

Suppose a subset of a metric space is dense. Does that imply that neighbourhoods of the elements of that dense subset covers the metric space. Is the converse true ? Can someone explain this ?
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$\mathbb R$ is not isometric with $\mathbb R^2$

show that $\mathbb R$ is not isometric with $\mathbb R^2$ (with the usual metrics). I want to use the first definition of continuity (i.e. the $\epsilon $ -$\delta$ stuff) but I don't see a way to ...
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A basic question on limit point of an infinite subset

Suppose in a metric space $X$ for a given $\delta >0$ we can find an infinite subset of $X$ in which the distance between any two elements is $\geq \delta$. Then can we say anything about whether ...
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Example of metric space that has more than two sets that are both closed and open?

I'm curious if there are examples of metric spaces having more than two sets that are both closed and open. Note: This is not for homework. This is to help me better understand the concepts of ...
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Prove that for every subset X of a metric space, Cl Cl X = Cl X.

Note: This is not homework help. This is simply to help me understand more about the topic of computational topology. Does anyone know how to prove this?
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need help with triangular inequality [duplicate]

I need to show that $||x|-|y||\le|x+y|\le|x|+|y|$. I proved the right inequality, $|x+y|\le|x|+|y|$, and now I need to prove the left inequality, $||x|-|y||\le|x+y|$. I though that I could do it by ...
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55 views

A basic question on metric space

Suppose In a metric space for every $x,y \in A$, for each $\epsilon$ we can find $x_0,y_0 \in B$ such that $$d(x,y) < \epsilon + d(x_0,y_0)$$ I want to prove that sup $d(x,y)_{x,y \in A}$ $\leq$ ...
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515 views

Prove that sequence space $\ell_p(\mathbb R)$ is separable

Problem: Prove that metric space $\left \langle \ell_p(\mathbb R), d_p(x,y)=(\sum_{i=1}^{\infty} |x_i|^p)^\frac{1}{p} \right \rangle$ is separable. Where $\ell_p(\mathbb R)=\left \{ ...