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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Function doesn't increase distance

Let $(X, d)$ be a metric space, $A \subseteq X$ with $A \neq \varnothing$, and $$f: A \rightarrow \mathbb{R}\quad\text{such that}\quad \left|f(x)-f(y)\right|\leq d(x,y),\ x,y\in A.\tag{$\ast$}$$ Let ...
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what would be a example in a general metric space about closedness and boundedness not imply sequential compactness?

Unlike in $R^n$, closed and bounded doesn't guarantee sequential compactness. Textbook examples includes sup metric and R^infinite metric. I am wondering what would be a example of closed and bounded ...
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79 views

Need to confirm: Sup Metric $C[0,1]$, question about boundary

For the sup metric, $C[0,1]$. Let $S \subset C[0,1]$ be given by: $$S=\left\{f:[0,1]\to \mathbb{R} \ : \ 0 \leq f\left(\frac{1}{2}\right)<1\right\}$$ The question is simple: is this set open or ...
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covering positively disjoint sets with finite balls

I have a question if $X$ is a separable bounded metric space and $A$,$B\subset X$ are sets such that $d(A,B)>0.$ Does there exists sets $A^{\prime}$ and $B^{\prime}$ such that each of them is a ...
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150 views

Continuity in metric space, TRUE or FALSE?

Let $(X,d)$ and $(Y,e)$ be metric spaces , and let $f: X \to Y$ be a function. True or false ? Give a proof or a counterexample as appropriate. $(a)$ If $d$ is the discrete metric on ...
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64 views

Group, metric, completion

Let $G$ be a group, $(G, \rho)$ - metric space, $p: G \rightarrow \mathbb{R}_+$ such that $p(x)=0 \iff x=e_G, \ \ p(x^{-1})=p(x), \ \ p(xy)\le p(x)+p(y), \ \ p(xy)=p(yx)$ Now let $\rho ...
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48 views

Function which doesn't increase distance

Let $(X, d)$ be a metric space, $A \subseteq X$ with $A \neq \varnothing$, and $$f: A \rightarrow \mathbb{R}\quad\text{such that}\quad \left|f(x)-f(y)\right|\leq d(x,y),\ x,y\in A.\tag{$\ast$}$$ Let ...
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357 views

Prove that if $Z\subseteq Y$, then $(g\circ f)^{-1}(Z)=f^{-1}(g^{-1}(Z)).$

Let $W ,X$ and $Y$ be three sets and let $f :W \to X$ and $g: X \to Y$ be two functions. Consider the composition $g \circ f: W \to Y $ which, as usual , is defined bt $(g\circ f)(w)=g(f(w))$ for ...
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61 views

Symmetric Operator with Different dot products

If I have a symmetric operator $A$ in a metric space $\mathscr{M}$. Then $\langle Au,v\rangle =\langle v,Au\rangle $ with the dot product defined in $\mathscr{M}$. My question is, if I keep the same ...
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192 views

Continuous map between metric spaces

Suppose $X,Y$ are metric spaces, let $A \subset X$ be a bounded subset of $X$ and $f: A \to Y$ to be a continuous bjection. Prove or disprove that $f^{-1}$ is continuous. Remark: If each closed ...
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77 views

Metric problems

I am taking the GRE in less than 10 days, and I have never taken analysis. And I would like to tackle metric problems and I was wondering if anyone could show me a certain strategy to solve problems ...
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63 views

Let $ f:(X, d) \mapsto (Y,d) $ be an mapping such that $ Graph (f) $ is connected. [duplicate]

Where $ X $ is connected. Does it imply $ f $ to be continuous?
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0answers
97 views

Regarding nowhere dense subsets and their measure.

A while ago it was made clear that a nowhere dense subset $P \subset [0;1]$ whose Lebesgue measure $\mu(P) = \mu([0;1]) = 1$ doesn't exist. But is it possible in principle to define a nowhere dense ...
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Is the set of integers Cauchy complete?

http://en.wikipedia.org/wiki/Complete_metric_space says that a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, ...
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Let $ f:(X, d)\mapsto (X, d ) $ be a mapping on compact metric space with $ d (f (x), f (y))<d (x,y) $for $ x\ne y $

I prove that $ f $ has a fixed point. My question is whether the point is unique and the mapping $ f $ is continuous.
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How to show that a continuous map on a compact metric space must fix some non-empty set.

Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$ I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
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77 views

Prove an inequality in $\mathbb{R}$

Let $ p,q \in \mathbb{R}, \; \lambda > 0, p \neq q$ (two points). For the two points $x_+, x_{-}$ with \begin{align*} x_+&=p+\lambda\cdot (q-p)\\ x_{-}&=p-\lambda \cdot (q-p) ...
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81 views

Is $(A \times B)^\epsilon \subseteq A^\epsilon \times B^\epsilon$?

While working on a problem related to my research, I had the following query. It pertains to product spaces: The Question: Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish (Complete separable metric) ...
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96 views

Isometry from Manhattan plane to Euclidean plane?

Does there exist an isometry from a Manhattan plane $A$ to a Euclidean plane $B$? I.e. a function $\varphi:A \to B$ that suffices $\|\varphi(a)\|_B = \|a\|_A$ for all $a \in A$, where $\| \cdot \|_A$ ...
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1answer
170 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
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Defining a metric space

I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches. Recently I've bought Topology by D. Kahn and am finding it ...
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440 views

Showing that the open ball is homeomorphic to $\mathbb{R}^n$ [duplicate]

I'm trying to prove that an open ball is homeomorphic to $\mathbb{R}^n$ but since this is the first proof of this kind that I try to give I'm having a little doubt on how to begin. I've had some ...
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Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
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Is a continuous function like a homomorphism/isomorphism for metric spaces?

If I had to define a notion of a homomorphism/isomorphism on metric spaces, I'd say something like this. Let $A$ and $B$ be metric spaces with norms $\| \cdot \|_A$ and $\| \cdot \|_B$ respectively. ...
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Cauchy sequence is convergent iff it has a convergent subsequence

Prove that if $\left ( x_{n} \right )$ is a Cauchy sequence in a metric space X then $\left ( x_{n} \right )$ is convergent if and only if $\left ( x_{n} \right )$ has a convergent subsequence. Note: ...
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Irrational P-adics

$\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete. I just want to be sure, are there p-adic rationals? If there are P-adic ...
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58 views

Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property

$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $. To prove ...
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If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.

A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$ Let $(X,d)$ be a metric space and let ...
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Is a 'normally' convergent sequence still convergent in a metric space which barely excludes its 'normal' limit?

For example, suppose $$ x_n = \frac 1n \\ X = (0, 1)$$ Is $x_n$ convergent in $X$? My guess would be no, since there exists no $x \in X$ which $x_n$ approaches; $x_n$ will eventually surpass any ...
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It is possible to generalize the “real” line to be able to embed $\omega_1$ or any uncountable ordinal into a finite segment of it?

This question is motivated from a previous question, but is in itself independent of it. So, I understand that it is not possible to embed $\omega_1$ or any uncountable ordinal into the real line, ...
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83 views

Metric Spaces needed for Differential Geometry

I've asked here about some texts about differential geometry which doesn't assumes that the reader knows general topology. I've got good references as Do Carmo's Differential Geometry of Curves and ...
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70 views

Equivalents metrics and closed sets

I have proven that if $ d $ and $ \rho $ are two equivalent metrics on a set $ E $ then these metrics define the same open sets in both metric spaces $ (E, d) $ as $ (E, \rho ) $. What I tried was ...
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A bijective mapping between metric spaces is open iff it is closed

Let $X$ and $D$ be metric spaces and suppose that $f: X \to D$ is one-one and onto. Show that $f$ is an open map iff $f$ is a closed map. how can I able to solve this problem
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239 views

Is the unit ball of a separable Banach space itself separable?

If $X$ is a separable Banach space, then do we know that its unit ball has a countably dense subset contained in the unit ball? This isn't obvious to me.
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Unit ball of a Separable Banach Spaces is metrizable

Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ...
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Let $(X,d)$ be a metric space and assume that $B_r^d(x)=B_s^d(y)$. is $r=s$ and $x=y$ [duplicate]

Let $(X,d)$ be a metric space and assume that $B_r^d(x)=B_s^d(y)$ where: $$B_r^d=\{ a \in X | d(a,x) < r\}$$ Now, is it always true that (a) $r=s$ (b) $x=y$ I made an elaborate argument on this ...
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100 views

Let $F$ be a subset of a metric space $(X, d)$ such that $\overline{F}$ is compact. Show that $F$ is totally bounded.

Let $F$ be a subset of a metric space $(X, d)$ such that $\overline{F}$ is compact. Show that $F$ is totally bounded. how can I able to solve this.I have no idea.thanks for your help
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How to prove boundary of a subset is closed in $X$?

Suppose $A\subseteq X$. Prove that the boundary $\partial A$ of $A$ is closed in $X$. My knowledge: $A^{\circ}$ is the interior $A^{\circ}\subseteq A \subseteq \overline{A}\subseteq X$ My proof ...
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$f$ continuous and surjective, $d_1(a,b)\le d_2(f(a),f(b))$, $X$ complete implies $Y$ complete

Let $(X,d_1)$ and $(Y,d_2)$ be metric spaces. Let $f : X \to Y$ be continuous and surjective. Suppose $d_1(a,b)\le d_2(f(a),f(b))$ for all $a,b\in X$. How can we show that if $X$ is complete then $Y$ ...
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Is the mapping $ d : X\times X \mapsto \mathbb {R} $ continuous?

Where $ (X, d) $ is a metric space. I want to prove it using sequential criteria. How do I tackle it?
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If 2 open balls define the same space, is it true that x=y and r=s?

Let $(X,d)$ be a non-empty metric sapce, $r$ and $s$ are postive radii, and $b_r^{d}(x)=b^d_s(y)$ for some $x,y \in X$. Is it true that $r=s$ ? Is it true that $x=y$? My answer would be ...
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A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement.

Let $E$ be a subset of a metric space $(S,d)$. Prove that: A point is in the boundary of $E$ if and only if it belongs to the closure of both $E$ and its complement. Here is what I thought: I'm ...
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Does the union of open neighborhoods of all points in a metric space cover the metric space?

Let $M$ be a metric space that is locally compact. Let $O_i \subset M$. Let $C$ be an open cover of $O_i$, and let $C'\subset C$. Define $U \subset O_i$ to be an open neighborhood of some $x\in O_i$ ...
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Is this function Lipschitz continuous?

Let $\mu \in \mathbb R^d$ be given. Is the function $f:\mathbb R^d \to \mathbb R^d$ defined as $f(x) := \exp(-\|x- \mu\|) (\mu - x)$ Lipschitz continuous? More specifically, for any $x, y \in ...
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Is the mapping that takes a metric to the induced intrinsic metric a closure operator?

To abbreviate the expression, "it holds that," I will write "iht." First a definition. Given a partially ordered set $(P,\geq)$, a closure operator on $P$ is a mapping $\mathrm{cl} : P \rightarrow P$ ...
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Can we extend any metric space to any larger set?

Let $(X,d)$ be metric space and $X\subset Y$. Can $d$ be extended to $Y^2$ so that $(Y,d)$ is a metric space? Edit: how about extending any $(\Bbb Z,d)$ to $(\Bbb R,d)$
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173 views

equivalence of compactness and countably compactness

Is there a way to prove that in metric spaces, compactness and countably compactness are equivalent, without using the Bolzano Weierstrass Property?
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68 views

find open balls $B_1,B_2,B_3,\ldots$ so: $U=\bigcup _{n\in \Bbb N} B_n$ , where $U=\{(x,y)\in \Bbb R^2 : y\gt x\}$

In the metric space $(\Bbb R^2,d_{\Bbb R^2})$: How can I find open balls $B_1,B_2,B_3,\ldots$ so: $U=\bigcup _{n\in \Bbb N} B_n$, where: $U=\{(x,y)\in \Bbb R^2 : y\gt x\}$. and why ...
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194 views

Show the $\operatorname{int}(A)$ is open.

So we want to show that the interior of any set $A$ is open. We will denote $\operatorname{int}(A)$ as the interior of $A$ which is the set of all interior points of $A$. I know in order to prove ...
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442 views

Intersection and union of complete subsets of a metric space

Does anyone know the two following proofs? (i) the intersection of any collection of complete subsets of metric space $(X, d)$ is complete. and (ii) the union of a finite number of complete ...