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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Discrete Sets in Certain Metric Spaces are Countable

I've recently started my first real analysis course, and we're studying metric spaces from Rudin's Principles of Mathematical Analysis (Baby Rudin). We have the following definition: Let $(X,d)$ be ...
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A question on Rudin's Book “ Principles of Mathematical Analysis”

On Theorem 2.27 (a), page 35, Rudin's proof is incomplete. If it is not the case $E' \subset E$ then the proof is false. Are you agree with this observation? (The theorem is: “If $X$ is a metric ...
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50 views

How to define derivative in Minkowski space

My understanding of derivative is like this: it is the unique linear mapping that sends the difference in $x$ to the difference in $f(x)$ when the difference in $x$ is small. To put it more ...
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Example of a continuous bijection on $\mathbb R^n$ whose inverse is not continuous [duplicate]

For $n \ge 2$ give example of a bijective continuous map $f: \mathbb R^n \to \mathbb R^n$ whose inverse is not continuous ; example of such a function is also an example of Does there exist a ...
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37 views

Let D be the usual Euclidean metric on M. Are the two identity functions $I_1$ and $I_2$ continuous functions where d is the Manhattan metric on M?

Let D be the usual Euclidean metric on M. Are the two identity functions $I_1:(M,D) \to (M,d)$ and $I_2:(M,d) \to (M,D)$ continuous functions where d is the Manhattan metric on M? The definition of ...
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Non-equivalent distances on $\Bbb{R}$

Let $\vert x-y\vert$ be the usual distance over $\Bbb{R}$ and $\gamma(x,y)=\Phi(d(x,y))$ where $\Phi(t)=\frac{t}{1+t}$. I would like to prove that the two distance are not equivalent. I now ...
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Metric Space : complete subset in a metric space [closed]

Proposition: Let (X,d) be a metric space, and let Y⊆X . (a) If Y is complete, then Y is closed. (b) If (X,d) is complete and Y is closed, then Y is complete. Use this proposition to argue that ...
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25 views

Proving p is a Metric on X

I have an exercise which I cannot get my head around. Essentially $X$ is a non-empty set and $p: X^2 \to \mathbb{R}$ satisfies (1) $0 \leq p(x,y) < +\infty$ (2) $p(x,y) = 0 \iff x=y$ and (3) ...
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16 views

Partition of bounded set on finite family of subsets with diameter less then 1

Let $(X,d)$ be a metric space with bounded metric $d$ which can take arbitrary small positive values. I wish to divide $X$ on the finite union of subsets with diameter $<1$? How to do it?
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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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Is it possible to show that the addition of two Cauchy sequences in $\mathbb R^n$ is also Cauchy for any metric?

A problem in my homework asks to show that the addition of two Cauchy sequences in $\mathbb R^n$ is also Cauchy. However, the metric is not specified. If we assume that we are dealing with the ...
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1answer
38 views

The formula for a distance between two point on Riemannian manifold

Given a Riemannian (say, connected) manifold $(M,g)$ one can produce the metric via a formula $d(x,y)=\inf_{\gamma}l(\gamma)$ where $\gamma$ is piecewise smooth curve joining $x$ and $y$. My question ...
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2answers
48 views

Example of closed, non bounded set in R^2

I am supposed to give an example of a closed set that is not bounded in $\mathbb{R}^2$. My idea was the graph of $y=1/x, \forall x$. If I take the complement of it, I get an open set. So the graph of ...
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2answers
59 views

Problem about metric spaces?

Let $X$ be an infinite set and let $d$ be the discrete metric on $X$. What sets in $X$ are open? Closed? Compact? Now, I know that $d$ will be either $0$ or $1$ since we are talking about the ...
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1answer
50 views

Let $X$ be a metric space. If $A ⊂ X$ is a compact set, prove that for any open covering, there exists a countable subcovering.

Let $X$ be a metric space. If $A ⊂ X$ has the property that every infinite subset of A has an accumulation point in $A$, show that for any open covering of $A$, there exists a countable subcovering. ...
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24 views

set of continuous functions to continuous functions: is $R$ complete?

Hi, I can do part (i) and (ii) but have trouble understanding part (iii). I can't intutively feel what the map R does. It takes continuous function to continuous function? How would I start the ...
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41 views

Topology, metric spaces, equivalence of metric spaces

The open $n$-cube is the set of all points $x=(x_1,x_2,\dots,x_n)\in\mathbb R^n$ such that $0<x_i<1$ for $i=1,2,\dots,n$. Prove that the open $n$-cube, considered as a subspace of $(\mathbb ...
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Could a complete metric space be a union of uncountably many nowhere dense subsets of it?

According to Baire's theorem, Any complete metric space can't be written as a union of a sequence of nowhere dense subsets of it. So, this assumes that the union is a union of countably many ...
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16 views

Euclidean Metric - $N=2$ Case

I have proven that the Euclidean metric is indeed metric, using Minkowski's Inequality. Now, my question is, using these properties to prove it for the general case of $\mathbb{R}^n$, can we simply ...
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Metric equivalence

Let $ (X,d_1)$ and $ (Y,d_2) $ be two metric spaces. Define a one to one function $ f : X\to Y $. Define a new metric on $ X$ as $ d'(x_1,x_2) = d_2(f(x_1),f(x_2)) $. Question 1) Are $ d_1 $ and $ ...
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1answer
24 views

Triangle Inequality Property for the Euclidean Metric

I've read in many of my books that the triangle inequality for a metric space of the Euclidean Metric is defined as: $$d(x,y) \leq d(x,z) + d(z,y)$$ But when I look up the proof, to help me ...
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43 views

Is closurness a necessary condition in cantor's interesction theorem?

Cantor's Intersection Theorem: Let $X$ be a complete metric space and let $\{F_n\}$ be a sequence of decreasing non-empty closed subsets of $X$. If $d(F_n)\rightarrow 0$ then $F=\bigcap_{n=1}^\infty ...
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Is the space $B([a,b])$ separable?

Let $a$, $b$ be two real numbers such that $a < b$, and let $B([a,b])$ denote the metric space consisting of all (real or complex-valued) functions $x=x(t)$, $y=y(t)$ that are bounded on the closed ...
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continues function statement in real analysis [closed]

I ran into a challenge, i read following sentence in one note. anyone could describe or prove it for me? F is a continues function at point $ x_0 \Leftrightarrow (x_n \to x_0 \Rightarrow ...
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Compact set example

Can you please give me an example of a set that is closed but not compact in R^2\Bbb? I know that a compact set is the one that is closed and bounded, and the set [a,b] is compact. But this question ...
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39 views

Prove the existence of disjoint open subsets

Let $A$ and $B$ be disjoint closed subsets of a metric space $(X,d)$. Give a direct proof for the existence of disjoint open subsets $U_a$ and $U_b$ of $X$ such that $A \subset U_a$ and $B \subset ...
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Proof about a subset of a metric space

Prove that a subset $A$ of metric subspace $(P, p')$ of metric space $(M, p)$ is open in subspace $(P, p')$, regarded as a metric space in its own right, if and only if there exists an open set $U$ in ...
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Fixed point in compact metric space

I guys! I try to solve the following small problem. However, I'm not able to prove the second part. In particular, I have some problems in using the compactness hypothesis on $X$ to find proper ...
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continuity of a identity map

Let $V_1=c[0,1]$ with $d_1(f,g)=max|f(t)-g(t)|$ and $V_2=c[0,1]$ with $d_2(f,g)=\int_0^1|f(t)-g(t)|dt$. Is identity map from $V_1$ to $V_2$ continuous? what about $V_2$ to $V_1$? I can't find what ...
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Characterization of Discrete Sets in R

Let A be a subset of $\Re$ . Does anyone have a characterization of discrete sets A ( which only have isolated points ) ? I'm coming up with A is discrete iff ( A is finite) or (A is infinite and ...
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Show that $\text{int}(\mathbb Q\times \mathbb Q)$ in $\mathbb R^2$ is the empty set and find the boundary of $\mathbb Q\times \mathbb Q$.

Question: Show that $\text{int}(\mathbb Q\times \mathbb Q)$ in $\mathbb R^2$ is the empty set and find the boundary of $\mathbb Q\times \mathbb Q$. Now that I read the questions correctly (thank ...
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confusion over how to show that f is continuous.

Okay , I'm just going to write down exactly what my book says and then ask my question. "Define the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ by $f(x_1,x_2)=x_1+x_2$. Prove that $f$ is ...
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Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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Limits of functions in metric spaces

My teacher said that in the definition of limit, the point in the domain, must be of accumulation, because otherwise the limit is not unique. Why? If the point is isolated, the function is continuous, ...
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Show that this f is continuous?

Okay , I'm just going to write down exactly what my book says and then ask my question. "Define the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ by $f(x_1,x_2)=x_1+x_2$. Prove that $f$ is ...
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Distance $\Psi(A,B)=\sup_{x\in E}\vert d_A(x)-d_B(x)\vert$ where $d_A(x)=\inf_{y\in A}d(x,y)$.

Let $(X,d)$ be a metric space, assume that $d$ is bounded. Denote $F$ the set of all closed set of $X$. Define $$\Psi(A,B)=\sup_{x\in X}\vert d_A(x)-d_B(x)\vert$$ where $d_A(x)=\inf_{y\in A}d(x,y)$. ...
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existence of unique fixed point

Let $(X,d)$ be a compact metric space and $f:X \to X$ satisfies $d(f(x), f(y))< d(x,y)$ for distinct $x$ and $y$. Then, show that $f$ has a unique fixed point. I tried this question by formulating ...
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continuity and closure questions - topology

Let $(X,d)$ be a metric space. Let $U \subseteq (X,d)$. let $k \in (X,d)$. Prove that if $U$ is fixed, $d(U,k)$ is a continuous function of $k$. Prove that $\overline{U} = U \cup V$ where $V$ is the ...
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What would be a standard framework, terminology, or procedure for extending function domains using isomorphisms?

Suppose we have an isomorphism $h:A\rightarrow B$ between spaces $A$ and $B$. Remark: Alternatively, we may consider a isomorphism $h:X(0)\rightarrow X(t)$, where spaces $X(t)$ are parametrized by ...
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Show that $d(u,v)=\exp(-\max\{j\ge 0, u_k=v_k \space\mbox{for}\space 0\le k\le j\})$ is a distance over $E=\Bbb{R}^\Bbb{N}$.

Let $E=\Bbb{R}^\Bbb{N}$, $u=(u_k)_{k\in\Bbb{N}}$ and $v=(v_k)_{k\in\Bbb{N}}$. Define $$ d(u,v) = \left\{ \begin{array}{ll} \exp(-V(u,v)) & \mbox{if}\quad u\ne v \\ 0 ...
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Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
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4answers
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How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
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Is my reasoning accurate?

$$\text{d}_{H}(A,B) = \max\left\{ \sup_{a\in A} \inf_{b\in B} \text{d}(a,b),\sup_{b\in B} \inf_{a\in A}\text{d}(a,b)\right\}$$ where $A$ and $B$ are two closed subsets of a metric space $(E,d)$ is a ...
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$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
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Let $(Y,\rho)$ be a metric space and $\rho : Y \times Y \rightarrow \mathbb{R}$ Prove that $\rho$ is a continuous function on $Y \times Y$.

Let $(Y,d)$ be a metric space and $d : Y \times Y \rightarrow \mathbb{R}$ Prove that $d$ is a continuous function on $Y \times Y$. I was thinking of the following : If $(a_{1},a_{2}) \in Y \times ...
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Showing $f$ is continuous on $M$ if $M=\bigcup_{n=1}^{\infty} U_n$

Let $f:(M,d)\to (N,\rho )$. If $M=\bigcup_{n=1}^{\infty} U_n$, where each $U_n$ is open, and if $f$ is continuous on each $U_n$, show that $f$ is continuous on $M$. Attempt: I note that ...
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58 views

Prove that this is a metric space?

I'm supposed to show that If X is the set of all functions on the interval $[a,b]$ and $\displaystyle d(f,g)= \int^{b}_{a}|f(x)-g(x)|dx\,$, then $(X, d)$ is a metric space. But I don't think it ...
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1answer
30 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
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1answer
19 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...
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1answer
37 views

Constructing a countable dense subset of a totally bounded set

Given a metric space $(X,d)$, and (non-empty) totally bounded set $E$ in $X$, is it possible to construct $D \subseteq E$ which is countable and dense? I feel that this should definitely be possible. ...