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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$f=(f_1,f_2,\ldots,f_n):X\longrightarrow Y_1\times Y_2\times\cdots \times Y_n$ is continuous $\iff f_i$ is continuous for all $i$

I have the following question for HW: let $(X,d)$ and $(Y_i,d_i)$ be metric spaces for each $1\leq i\leq n$. Suppose that for each $1\leq i\leq n$, $f_i :X\longrightarrow Y_i$ is ...
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Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
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Find a metric on the simplex so that every transposed positive stochastic matrices becomes a contraction.

A stochastic matrix $P$ is a $n \times n-$matrix with entries $p_{ij} \in [0,1]$ so that $\sum_{k=1}^n p_{ik} = 1$ for every $i \in \{1,...,n\}$. The matrix $P$ is called positive, if no entry ...
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Understanding a theorem of Dynamical systems, mostly definitions

I am reading, the following paper for my research: http://www.sciencedirect.com/science/article/pii/S0022247X00973438 I need to know what are the precise definitions of the following terms, since I ...
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Proving that a set A in a metric space is bounded iff it belongs to a ball with radius r>0

Let $(X,d)$ be a metric space. Prove that $A\subset X$ is bounded if and only if $A\subset B_r (a)$ for some $a\in X$ and $r>0.$ This is my own proof to the question. Let $B_r(a)=\{x\in X : ...
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Image of a precompact under the action of uniformly continuous function is a precompact

Suppose we have two metric spaces $(X, \rho_x)$ and $(Y, \rho_y)$ and a uniformly continuous function $f\colon X \to Y$. The problem is to prove that image $f(A)$ of every precompact $A \subset X$ ...
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Find a metric on $\mathbb{R}$ with the property that the sequence of natural numbers is Cauchy.

I'm trying to solve the following question: Find a metric $d$ on $\mathbb{R}$ that is equivalent to the usual metric and has the property that the sequence $(n)_{n=1}^{\infty}$ is a Cauchy sequence. ...
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Problem 13 chapter 4 from baby Rudin

Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous real function defined on $E$. Prove that $f$ has a continuous extension from $E$ to $X$. Could the range space ...
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Is there a continuum contained in a compact metric space?

I am reading a paper by M A Armstrong called "the fundamental group of the orbit space of a discontinuous group". In it, he refers to a theorem which says - any light open map between compact metric ...
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The cartesian product $M\times N$ is complete if, and only if $M$ and $N$ are complete.

The cartesian product $M\times N$ is complete if, and only if $M$ and $N$ are complete. My approach: Let $M$ and $N$, complete metric space, then we take a cauchy sequence ...
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completion of a complete metric space is itself

Suppose $(X,d)$ is a complete metric space and $(Y,d)$ is its completion . What can be said from this $?$. I think $Y=X$ is the answer . Am I correct $?$
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Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
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Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \begin{equation} \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, ...
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Proof that the closed interval in $\mathbb{R}$ is connected

Let $C$ be an open and closed subset of $[a,b]$. WLOG, assume $a \in C$. Set $A = \{x \in [a,b]: [a,x] \subseteq C\}$. Since $a \in A$, sup$A$ exists. Let $\epsilon > 0$. Then, (from real ...
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What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
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Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$ \int a(x,y)f(y)\mathrm dy = b(x)\tag{1} $$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
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Ball with euclidean metric - mistake in book?

In $\mathbb{R}^2$, the ball with euclidean metric $d_{l_2}$ is defined, in terence tao's analysis vol II, as: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
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Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
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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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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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Can a metric subspace be completely covered by balls after a finite number of steps?

Let $X$ me a metric space with distance $d$ and $A$ be a subspace of $X$. Let $B_\varepsilon(x)$ be the open ball centered in $x$ with radius $\varepsilon$, i.e. $\{y\in X\mid d(x,y) < ...
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Takahashi convex metric spaces

A Takahashi convex metric space is a metric space $(X,d)$ such that $\exists W : X \times X \times [0,1] \rightarrow X$ that satisfies : $d (u, W(x,y; \lambda)) \leq \lambda d(u,x) + (1- \lambda) ...
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The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
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Given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected.

Let $X$ be a (metric) space such that given any two points $x, y \in X$ there exists connected set $A$ such that $x, y \in A$. Then $X$ is connected. Let us consider a continuous function $f : X \to ...
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Relationsip between two definitions of the christoffel symbol?

When I first started learing about tensor calculus, the professor defined the Christoffel symbol as $$\Gamma ^a _{bc} = Z^a \cdot \partial_b Z_c $$ Where $Z^a$ is a contravariant basis vector and ...
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$A \subset \Bbb R$ such that $A$, $clA$, $int(A)$, $cl(int(A))$, $int(clA)$ are pairwise distinct

Do there exist subsets with internal closures $A$ of $\mathbb R$ such that $A$ , $\bar A$ , $A^\circ$ , $(\bar A)^\circ$ , $\overline{A^\circ}$ are pairwise distinct? I found an example from a book ...
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A metric space of which the geodesic is not a metric

The text book in my course has an exercise about finding a metric space whose (usual) length metric is not a metric. It wants me to find a metric space $(X,d)$ satisfying $d'(x,y)=0 \ \ $for some ...
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A set $U$ is open iff it is union of open balls

Let $(X,d)$ be a metric space. Consider the collection $\mathcal{T} = \{ U \subset X: \forall u \in U, \exists r>0 \; \; , B_r(u) \subset U \}$. We showed that $(X, \mathcal{T} )$ is a topological ...
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Pointwise convergence imply uniform convergence

I am trying to find a condition under which a sequence of continuous functions on a metric space (or more generally in a topological space) which point wise converge to some function f should imply ...
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How to show that metrics generate the same topology?

Let $(X, d)$ be a metric space, let $c$ be a positive real number, and define a new metric $d'$ on $X$ by $d'(x,y) = c \cdot d(x,y)$. Prove that $d$ and $d'$ generate the same topology on $X$. ...
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If $E_i$ is open show $\cap E_i$ is open

Question If $E_i \subseteq \mathbb{R}^p$ is open for all $i=1,2 \dots, n$. Show that $\displaystyle \bigcap_{i=1} ^n E_i$ is open. My attempt: Let $x \in \displaystyle \bigcap_{i =1}^n ...
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Measuring dispersion

I am trying to define a proper metric for characterizing dispersion of a set of $k \in \mathbb N$ points distributed over different spatial grids. Formally, given different 2-dimensional grids ...
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Proof for Norms in Vector Spaces

Prove that if a norm $\|x\|$ on a real vector space satisfies the parallelogram law, then the polarization identity defines an inner product and that the norm associated with this inner product is the ...
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Proving the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem

I need to prove the Urysohn's metrization theorem by using the Nagata-Smirnov's metrization theorem. Urysohn's metrization theorem: Every regular second-countable topological space is ...
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Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
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Density character of a metric subspaces

Is it true that if $M$ is a metric space and $N$ is a metric subspace of $M$ (I mean, $N\subseteq M$ and the metric defined on $N$ is the same metric on $M$ restricted to $N$) then the density ...
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Proof of Kuratowski-Wojdyslawski theorem

I was reading the Wikipedia page on Kuratowski Embedding, and the following result is stated: The Kuratowski–Wojdysławski theorem states that every bounded metric space $X$ is isometric to a ...
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At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
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How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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non-separable metric space and measurablility of its elements

I'm studying Skorokhod space, which consists of cadlag functions, and I encountered the following sentence: If a metric space $(\mathbb{S}, \mathcal{S}, d)$ is not separable, then functions that ...
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Physical Meaning of Minkowski Distance when p > 2

Suppose we have two vectors in $u, v \in \mathbb{R}^d$. For $p \geq 1$, the Minkowski distance between these vectors is defined as $ \lVert u - v \rVert_p = \Bigl( \sum_{i=1}^d \lvert u_i - v_i ...
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check proof that $B[a,b]$ is not seperable

This is what I have to prove: prove that metric space $B[a,b]$, $a<b$ is not separable. Where $B[a,b]$ is the set of all bounded and defined functions on $[a,b]$, with the metric ...
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“Limit set” of infinite measure for a “Cauchy” sequence

Let $\{A_n\}$ be a sequence of sets $A_n\subset X$ of finite Lebesgue measure $\mu$ with the property that$$\forall\varepsilon>0\quad\exists N\in\mathbb{N}^+:\forall n,m\geq N\quad\mu(A_n\triangle ...
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Metric space with measure and a special property

Let $R$ be a metric space endowed with a (complete) measure $\mu$ satisfying the following condition: all the open and closed sets of $R$ are measurable and for any measurable set $M\subset R$ and any ...
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Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

This question is related to this one: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set Kuratowski's embedding indeed seems to be the ...
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Completely metrizable space, complete norm

Let $E$ be a normed, completely metrizable space. Prove that the initial norm is complete. How can I go about solving this problem? I will be grateful for all your hints. Thank you!
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In regards to metric spaces, does $d^\star$ have an accepted name, or notation? Do any authors use it?

(I write $\omega$ for the set $\{0,1,2,\ldots\}$.) Let $X$ denote a metric space with metric $d$. Define a function $d^{\star} : X^\omega \times X^\omega \rightarrow [0,\infty]^\omega$ by writing ...
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Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
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Conditions for Metricization of Cartesian Product of Metric Spaces

Let $M_1$ and $M_2$ be metric spaces with metrics $\rho_1$ and $\rho_2$ respectively. What are some necessary and sufficient conditions on $f:\mathbb{R}_{+}^2\to\mathbb{R}_{+}$ that make ...