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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I want to prove that $f$ is continuous if its graph is closed

This is an exercise from Rudin's 'Functional Analysis': Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph $f:X\to K$ is a closed subset of $X\times K$. Prove that ...
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Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
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distance between a real and R\Q

Please how to prove that $d(x, R\setminus Q)=0, d(x,Q)=0$ for all $x\in \mathbb{R}$ ? I know that $d(x,R\setminus Q)=\inf_{a\in R\setminus Q} d(x, a)$ but how to continue ? Can i say that As ...
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Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n (q_k-p_k)^2]^{1\over 2}$.

Let $p=(p_1,p_2,...,p_n)$ and $q=(q_1,q_2,...,q_n)$ be points in $\mathbb{R^n}$ with $p_k<q_k$ for each $k$. Let $R=[p_1,q_1]\times \cdots\times[p_n,q_n]$ and show that diam $R=d(p,q)=[\sum_{k=1}^n ...
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Zero-dimensional separable metric spaces

I have to prove that every separable metric space, which is zero-dimensional is isomorphic to a closed subset of the Baire space. Maybe I can use the Baire category theorem, but I don't know how.
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How to show $dist(p,S) = 0$, then there exists a sequence in $S$ converging to $p$

Let $S \subset (M, d)$, where $(M,d)$ is a metric space Let $dist(p,S) \equiv \inf\{d(p,s) | s \in S, p \in M\}$ I wish to show that if $dist(p,S) = 0$, then there exists a $(p_n)$ in $S$ converging ...
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Type of convergence of a Cauchy sequence of functions on a complete metric space?

Let $\{f_n\}$ be a Cauchy sequence of functions defined on a complete metric space $E$. Then $f_n \to f$ on $E$. What is the type of this convergence? Is it pointwise?
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Let $f$ be a function $f:[0,1] \to [0,1] \times [0,1]$ now can we find $f$ with following conditions?

Let $f$ is a function $f :[0,1] \to [0,1] ×[0,1] $ now can we find $f$ with following conditions ?: 1- f be continues and one to one . 2- f be continues and onto . 3- f be continues and one to one ...
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Is a Normed Vector Space Necessary to Prove Path Connectedness?

Path connectedness seems to be defined in a topological space, but can the existence of a path be proven without using the functions of vector addition, scalar multiplication and norm ? For example, ...
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Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$

Let $(M,d)$ be a metric space and define: $d' : M$x$M \rightarrow R$ Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$ I know that $d'$ defines a metric on M, since d is a ...
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Does nonexpansive property in H-norm imply nonexpansive in 2-norm?

Suppose $\|f(x) - f(y)\|_H \le \|x - y\|_H$. In other words, $f$ is nonexpansive in the norm with respect to positive definite H: $\|z\|_H = z^T H z$. Can we then say something along these lines: ...
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Is the set of Darboux integrable function a metric space under the given distance definition?

Let $X$ be the set of all Darboux integrable functions in the domain $[0,1]$. Distance between two function is defined to be $$d(f,g) = \int_0^1|f-g|\,dx.$$ Is this a metric space? I ...
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Am I making some mistake in proving that $S$ is dense subset of $C[0,1]$?

Consider the space $X=C[0,1]$ with its usual 'sup-norm' topology.Let $$S= \{ f \in X : \int_{0}^{1} f(t) dt \neq 0\}$$ Show that $S$ is dense in $X$ We note that convergence with sup norm is ...
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Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n\times n$ matrices with real entries

Show that the function $A →\|A\|$ defined by $\sup \|Ax\|/\|x\|$ is a norm in the space $M_n$ of $n \times n$ matrices with real entries. Definition 1.26. Let $X$ be a linear space (over $R$). A ...
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Retracts and dense $G_{\delta}$ sets

Suppose $X$ is a complete separable metric space and $Y\subset X$ is a retract of $X$ with retraction $r\colon X\longrightarrow Y$. I am interested in the following question: Given a dense ...
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Is the metric on the circle, induced from the plane, not a flat one?

My question concerns the highlighted part posted below, from Wikipedia article. (Link to the revision at the time of this post.) I'd say I can't detect the curvature of the unit circle if I go along ...
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Get a bounded metric from a metric - triangle inequality for $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ [duplicate]

This is related to Proof that every metric space is homeomorphic to a bounded metric space but I can remember that if $d$ is a metric, then $d'(x,y):=\frac{d(x,y)}{1+d(x,y)}$ is also a metric that ...
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Non-separability of normed spaces

I would like some hints to decide when a normed space is separable or not. I really understood the definition and the classic examples of separable spaces but when I go to show that a space is ...
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To construct a continuous map on compact metric spaces (cf. Gromov-Hausdorff Limit)

This is about Gromov Hausdorff limit on compact metric spaces (Reference A course in metric geometry - Burago Burago and Ivanov, 268p. EXE 7.5.8) Definition : $d_{GH}(X,Y)<\epsilon $ if there ...
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Compactness Theorem (Propositional Logic) and Compactness (Metric spaces). [duplicate]

Definition. A subset $E$ of a metric space $(X,\tau)$ is compact if every open cover of $E$ has a finite subcover. Theorem (Compactness Theorem). A set $\Gamma$ of formulas is ...
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infinite subset of discrete metric space is not compact

The question is Im not really sure how to go about this So far i am trying to show that for an open cover of the infinite subset X, there isn't a finite sub cover and therefore X is not compact I ...
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Metric spaces and normed vector spaces

Studying I learned that there are some theorems and definitions that need a metric structure on the space in which we are working, for example the definition of local maximum needs a metric space or ...
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Why pseudo-Riemannian metric cannot define a topology?

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. Does this imply that in cosmology, say through FLRW metric, ...
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Meaning of amalgamated metric sum of $A_n$’s over $0$ and $d_n$ inherited from $\mathbb{R}^2$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
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What can you say about interior points of a non empty subset of real numbers?

Given that A is a non-empty subset of real numbers, if I(A) denotes the set of interior points of A; then I (A) is:- a) empty. b) singleton. c) a finite set containing more than one element. d) ...
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How calculate with Riemannian metrics (e.g. Multiplication and Divison)?

I have no idea how to handle the following Riemannian metrics, how to find the estimates for the bound and how to actually calculate with $g$ and $d$. Do I need to use the matrix representation? Or ...
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Proving that a set is closed with respect to a defined metric

Let $M = [0,1]^{[0,1]}$ Prove that the set of increasing functions $$ J := \{f \in M : \forall \space a,b \in [0,1], a \leq b : f(b) − f(a) \geq 0 \} $$ is a $d$-closed subset of $M$ where ...
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compact set in a space of functions continuous in $\mathbb{R}$

As it is known the set $\mathcal{B}=\{ f:\mathbb{R}\rightarrow \mathbb{R} : f \mbox{ is continuous}\}$ it is not a metric space with the metric $d(f,g)=sup_{x\in \mathbb{R}}\| f(x)-g(x)\|$ it can ...
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Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space?

Suppose $<M,d>$ is a metric space. Does lim$_{a \rightarrow b } \space d(a,b) = 0 $ imply completeness in a metric space? Or maybe lim$_{a \rightarrow b } \space d(a,b) \neq 0 $ implies ...
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Show that mapping is a contraction?

Show that the mapping $f:\Bbb R \to \Bbb R $, $f(x)=1-xe^x$ is a contraction. I tried everything i could think of but i cant get it to work. Witch is not much since i couldn't really find any ...
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Show that the collection of open balls in two metric spaces are identical

I am having trouble trying to prove the following statement. I can see why it would be true intuitively, however, I am having trouble formalising the proof as the notation is quite confusing. Show ...
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Limit of Riemannian manifolds is not Riemannian

I want to prove that $D$, standard unit ball in ${\bf R}^2$ with $|\ |$, with a metric $\| \ \|_1$ is a limit of Riemannian manifolds $X_i$. Here problem is to find $X_i$ (If necessary, all metrics ...
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The name of a polygon defined by multiple overlapping annuli

I am working on a problem in a metric space where points are partitioned into various annuli. If there exists multiple annuli that define a set of points then a polygon can be formed from their ...
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lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$ in Metric Space - Implications

A Metric Space $<M,d>$ is given by the Metric $M$ and distance function $d$ If there exists a Cauchy Sequence $x_n$ such that: lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$, for some $a \in ...
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How to determine the limit of a sequence in a metric space

If I'm trying to prove that a Metric space $(M,d)$ is not complete I have to find a Cauchy a sequence that doesn't converge in $M$. Using the following Metric Space as an example: $M = \{ x \in ...
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In Pugh's analysis book, why are these metric spaces?

Pugh introduces the notion of metric space in chapter 2 as follows Definition: A metric space is a set $X$ equipped with a metric $d$ Clear! For example, a metric space is $(\mathbb{R}, ...
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What does it mean exactly by a metric “generates” a topology?

For example, the discrete metric $d(x,y)$ where $d(x,y) = 1$ if $x\neq y$, $d(x,y) = 0$ if $x = y$ "generates" the discete topology $\tau$ where $\tau = 2^X$ Can someone clarify exactly what is meant ...
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On preimage of open sets of functions on real line having at most countably many discontinuity points

Let $f:\mathbb R \to \mathbb R$ be a function whose set of discontinuity points is at most countable ; is it true that for every open set $G \subseteq \mathbb R$ , there is an open set $U$ and a ...
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Showing that the following two metric are equivalent.

Let $(X,d)$ be a compact metric space, and $f \colon (X,d)\to \mathbb{R}$ is a continuous function such that if $x,y \in X$ and $x \neq y$ than $f(x) \neq f(y)$. Let $t \colon X \times X \to ...
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what are the geodesics in the hyperbolic upper half plane?

In the upper half-plane $$ H = \{(x, y) \in \mathbb{R}^2 \mid y > 0\} $$ the distance between the two points (a,A) and (b,B) is set by the shortest curvature in metrix $$ F(y) = \int_a^b ...
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Find a metric $d$ on $X$ such that $(X, τ^{(d/X)})$ is not connected

$X = (\{0\} \times [-1,1] \cup \{(x,\sin(π/x)) : x \in (0,1]\} \subset \mathbb{R}^2$ Find a metric $d$ on $X$ such that $(X, τ^{(d/X)})$ is not connected. Note: $τ^{(d/X)}$ denotes the metric ...
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Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in ...
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Metric space with two similar points which are not in the same orbit.

Is there an example of a metric space $X$ with two points $p$ and $q$ so that for every $r>0$ the ball with radius $r$ and center $p$ is isometric to the ball with radius $r$ and center $q$ and yet ...
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How can I prove that

Let we have the following ultrametric space $(z,|.|_2)$ such that if $x=r.2^n$ then $|x|_2=2^{-n}$ how can I prove that the topology produced by this metric isn't discrete topology ?
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If we think of infinity as a number, how does it affect the compactness/completeness of a metric space?

I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. ...
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A metric on the natural numbers

Does there exist a complete metric on the set of natural numbers such that $\{n,\,n+1,\,n+2,\,\cdots\}$ is a closed ball for each $n$?
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Mahalanobis distance

Suppose there is a function $f$, for which we know the inequality $$f(r)\leq r$$ is true, where $r=||x-y||_2=\sqrt{(x-y)^T (x-y)}$ is the Euclidean distance. If now we use the Mahalanobis distance ...
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Verification of proof of continuity between metric spaces and deduction from proof

Let $M = [0,1]^{[0,1]}$ and $d(f,g) = \sup{\{\lvert f(x) - g(x)\rvert \mid x \in [0,1]\}}$. For $a,b \in [0,1]$ let $\phi_{a,b}(f) = f(b) - f(a)$ ($\phi$ maps from $M \to \Bbb{R}$). Assume that ...
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compact metric spaces and infimum

I am currently revising metric spaces and have come across a question which I am unable to answer and have no idea how to begin with. Let $(M,d)$ be a compact metric space. Suppose $T \colon M \to ...
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General approach to determine completeness of metric space

I've looked at a few questions online asking to determine the completeness of Metric Spaces. 2 such examples of metric spaces $(M,d)$: 1) $M = \{ (x,y) \in \mathbb{R}^2 \space : y>0 $ or $ x=0=y ...