Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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Closed subsets of empty interior are of 1st category

In a metric space is it true that closed sets with empty interior are of 1st category? I.e., that it can be represented as a at most countable union of meager sets? Thanks
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Is point-to-set distance function $C^\infty$ for $\mathbb{R}^n$

Let $x\in \mathbb{R}^n$ and $Q\subset \mathbb{R}^n$. Then we define the point-to-set distance function as: $$ d_Q(x) = \inf_{y \in Q} \| x-y\| $$ It's continuous for every normal space (not only $\...
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Hölder's inequality/Cauchy-Schwarz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) \...
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Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
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Prove that $\delta$ is a metric in $\mathcal{K}(X)$

Let $(X,d)$ be a complete metric space. We define $\mathcal{K}(X)=\{K \subset X : K \text{ is compact and non empty}\}$ Define $d'(A,B)=sup_{a \in A}\{d(a,B)\}$ Show that $\delta$ ...
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how to prove the sequence based definition of a closure in metric spaces

Let $A$ be a subset of a metric space $\Omega$. By definition, the closure of $A$ is the smallest closed set that contains $A$. How to prove that alternativelly, the closure is given by (1): $\bar A =...
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Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
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Does it make sense to apply the Manhattan metric to an arbitrary graph?

I overheard someone talking about using the Manhattan metric against nodes on an arbitrary graph (or even a tree). At the time, I didn't think much of it, but having dwelled upon it, does it even make ...
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Confused about proof that diameter of a closure of a set is the same as the diameter of the set.

Definition Let $E$ be a nonempty subset of a metric space $X$, and let $S$ be the set of all real numbers of the form $d(p,q)$, with $p \in E$ and $q \in E$. The supremum of $S$ is called the diameter ...
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Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...
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Show that the closed ball $B[x,1]$ in $c_0$ is not compact.

Consider $c_0 = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}:a_n \to 0\}$. Show that the closed ball $B[x,1]=\{y \in c_0 : d_{\infty}(x,y)\leq 1\}$ is not compact in $c_0$. Were $d_{\infty}(x,y)=sup_{...
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25 views

Question about disconnected metric spaces

The definition of disconnectedness that I've been taught is that a metric space $(X,d)$ is disconnected if there exists two non-empty disjoint open sets $A$ and $B$ such that $X=A\cup B$. My ...
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Classification of Proper Maps between domains in $\mathbb{R}^n$

Suppose $f:D_1\to D_2$ is a continuous map between domains in $\mathbb{R}^n$. Show that $f$ is proper iff for every sequence $(x_n)$ in $D_1$ which accumulates only on $\partial D_1\cup\{\infty\}$, ...
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Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$

Let $(X,d)$ be a metric space. Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$. If $F\subseteq X$ is closed then $K\subseteq X$ compact implies $K$ ...
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p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
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Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
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On a condition when bounded sets in $\mathbb R^n$ is convex ?

Is it true that a bounded set in $\mathbb R^n$ , $n>1$ , is convex iff every straight line through an arbitrary interior point of the set intersects the boundary of the set in exactly two points ? ...
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64 views

Connected-ness of the boundary of convex sets in $\mathbb R^n$ , $n>1$ , under additional assumptions of the convex set being compact or bounded

Is the boundary of every compact convex set in $\mathbb R^n$ , ($n>1 $ ) connected ? is it path connected ? What if we assume only that the convex set is bounded , is the boundary connected ( and ...
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If the boundary of a convex set in $\mathbb R^n$ ($n>1$) is connected , is it necessarily also path-connected ?

If the boundary of a convex set in $\mathbb R^n$ ( where $n>1$) is connected , is it necessarily also path-connected ?
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$\epsilon$-dense subsets on $\mathbb R/\mathbb Z$.

Let $\langle M, d\rangle$ be a metric space. We say that $A \subset M$ is $\epsilon$-dense if every open ball of radius $\epsilon$ contains a point of $A$. Now let $T=\mathbb R/\mathbb Z$, the ...
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Sufficient conditions for embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.

This is a followup to a question I asked in this thread. I'm posting separately so points can be awarded. Hopefully someone can help me with a reference for this problem, or the construction. I ...
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On visualizing the spaces $\Bbb{S}_{++}^n$ and $\Bbb{R}^n\times\Bbb{S}_{++}^n$ for $n=1,2,\ldots$

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive-definite $n\times n$ real matrices. I am looking for hints concerning the visualization of such spaces for $n=1,2,\ldots$. I know that $\Bbb{...
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Two exercises about hypermetric spaces

Take $S$ to be the collection of all subsets of $\{1,\dots,n\}$. If $x, y$ are in $S$, define $d(x,y)$ as the number of elements of the symmetric difference $x\triangle y$. Exercise 2.1. Show ...
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Embedding a set of $n$ points with a given metric in $\mathbb{R}^n$.

Hopefully someone can help me with a reference for this problem, or the construction. I have a metric defined on $n$ points in $\mathbb{R}^2$. Is it possible to find a higher dimensional Euclidean ...
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Any compact metric space is Borel equivalent to some subset of $[0, 1]$

In Petersen's Riemannian Geometry book I encountered the following statement : Any compact metric space $X$ is Borel equivalent to some $S \subset [0, 1]$ i.e. there is a bijection $f : X \rightarrow ...
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Prove that the convergent sum of a real sequence is a metric

I want to show that $$ \varrho(\{a_n\},\{b_n\})=\left(\sum_{n=0}^\infty{(a_n-b_n)^2}\right)^{1/2} $$ is a metric, where $\{a_n\}_{n\in\Bbb N}\in \ell_2$, and $\ell_2$ is the set of all real sequences ...
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Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ?

Let $A$ be a convex set in $\mathbb R^n $ , where $n>1$ ; then is it true that $\bar A \setminus A^{\circ}$ i.e. the boundary of $A$ is connected ? if yes , then is it also path connected ?
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In $\mathbb Q_p$, proving every open ball is the disjoint union of more than one open ball

I'm reading the Foundations chapter of Gouvea's p-adic Numbers: An Introduction, and I'm trying to solve the following problem he poses to the reader: Take the $p-$adic absolute value on $\mathbb ...
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Show that if $X$ is sequentially compact, then $X$ is complete and totally bounded

Given a metric space $X$ which is sequentially compact (i.e every sequence has a converging subsequence), show that $X$ is complete and totally bounded. I've already shown that $X$ is complete, since ...
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A question on metric spaces which does not have Lebesgue covering property

Let $X$ be a metric space , $\{U_{\alpha}\}$ be an open cover of $X$ which has no Lebesgue number . So for every $r>0 $ , there is an open ball of radius $r$ which is not contained in any open set ...
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Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that $B(x,\...
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Prove that $G(f)$ is homeomorphic to $X$. [duplicate]

Let $X,d$ be a metric space .Let $f:X\to \mathbb R$ be a continuous function.Define $G(f)=\{(x,f(x)):x\in X\}$. Prove that $G(f)$ is homeomorphic to $X$. My try: Since $f$ is continuous then $G(f)$ ...
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Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow Y$....
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Little confusion about connectedness

Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$. So, $\overline X$ is also connected , as closure of connected set ...
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Find the limit points and exterior points of the following

Let $X=\mathbb R$, with the usual metric on $\mathbb R$ and $A=((0,1)\cap \mathbb Q)\cup$ {$2,3$}. Find the limit points of $A$, exterior points of $A$, $A^o$, $\overline A$ and $\partial A$. Can ...
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Proving that the ball is converx

I need to prove that the ball $B(x,r)=\{y\in \mathbb{R^n}:||y-x||<r\}$ is convex. How to do this?
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50 views

What exactly is the distance of two elements in $C[0,1]$?

If $C[0,1]$ — the set of all continuous functions from $[0,1] \rightarrow \mathbb R$ — is equipped with the metric $||\cdot||_1$ (1-Norm), then what is the distance between ...
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Bounded complete metric space is compact?

This question may seem trivial, but in topology we were taught that in a complete metric space, a subset of that space was compact if and only if it is closed and bounded. Moreover, we are told that ...
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Is $ \text{Int} \overline{B(a;r)} = B(a;r)$ for a metric space $(X,d)$?

I think this is true in general. To give a brief outline of a proof: Let $ \text{Int} \overline{B(a;r)} = U $, I claim that if $a \in U \implies a \notin Fr(B(a;r))$ so $a \in \text{Int}B(a;r)$ ...
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Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
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If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Let $X$ be a metric space such that every continuous function $f:X \to \mathbb R$ is uniformly continuous ( here $\mathbb R$ is equipped with the standard euclidean metric ) , then is it true that for ...
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If $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected.

I'm trying to show that if $\overline A\cap B = A\cap \overline B = \varnothing$, $A\cup B$ is disconnected. First of all, I think I have to assume that $A$ and $B$ are nonempty, or else the statement ...
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A subspace of a metric space is normal

Is it true that every subset $Y$ of a metric space $X$ is a normal topological space? I think the answer is yes, because $Y$ is a metric subspace of $X$ equipped with the induced metric by the one of ...
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Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
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Normed space of bounded functions $f:\mathbb{N}\to\mathbb{N}$

Let $X = \{f:\mathbb{N}\to\mathbb{N}: \exists M\in\mathbb{N} \forall n\in\mathbb{N} f(n) \leq M\}$. Define a norm on $X$ by defining for $f\in X$: $$||f|| = \sum_{n=1}^\infty \frac{f(n)}{2^n}.$$ Is ...
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Prove that a finite union of closed sets is also closed (using limit points)

Let $F_i$ be a family of closed sets, then we know that $\bigcup_{i=1}^nF_i$ is closed. Proving that statement is equivalent to proving: If $p$ is a limit point of $\bigcup_{i=1}^nF_i$ then $p\in\...
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If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact?

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again $X$...
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Lower semicontinuity on a metric space

I'm trying to prove something about lower semicontinuity for a map on a metric space $(X,d)$. I will try to write here my idea of the proof, hope someone can approve or contest it. Def. Let $(X,d)$ ...
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Dense-in-itself open sets in a subspace of the real line

Given an uncountable set $X\subset [0,1]$ it is easy to write $X$ as a disjoint union of a perfect set $P$ (perfect in the subspace $X$) and an at most countable set $C$: just take $P$ as the set of ...
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If $f:X \to [0,1]$ be an onto continuous map and $\{f^{-1} (y)\}$ is compact then Is $X$ compact?

If $f:X \to [0,1]$ is an onto continuous map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again $X$ is ...