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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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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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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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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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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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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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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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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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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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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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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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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101 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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30 views

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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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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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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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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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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On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
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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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156 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. ...
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proof that a set of all bounded real valued functions is complete.

I am trying to understand the proof below. I know that a set A is complete if all Cauchy sequences converges in A. I don't understand 7th line of the proof. Why do we consider particular $x_0 \in X$ ...
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$\partial(S') \subset \partial S$ iff $S' \cap S^o \subset (S')^o$

Usually I can come up with some ideas but this time I don't. It would be great if you can tell me how I would make use of the first part of the question to prove the equivalent relation. Question: ...
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Definition of a metric space: why $E\times E\rightarrow\mathbb{R}$?

In the definition of a metric space Let $E$ be a set and $d:E\times E\rightarrow\mathbb{R}$ be a function. $d$ is a distance on $E$ if ..., why is the function $d:E\times ...
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Distance to a closed set is continuous.

I want to prove that given a metric space $(M,d)$ and $F \subset M$, then the function $f_F: M \to \Bbb R$ given by $f_F(x) = d(x,F) = \inf\{d(x,y) \ : \ y \in M\}$ is continuous. Take $x \in M$. If ...
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Extend Metric Space Challenge

Let $(E, D)$ be a metric space. Consider $D_1: E\times E \to \mathbb{R}$ where $$ D_1(x,y)=\frac{D(x,y)}{1+ D(x,y)}. $$ I read some note about it but I want to find why $D_1$ is also a metric and ...
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Is there always an equivalent metric which is not complete?

I have seen that completeness is not a topological property like compactness or connectedness. I have seen some examples also showing that there are two equivalent metrics one of which is complete and ...
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Cauchy sequence and metrics

I'm having trouble with another analysis homework problem: Let $x_n$ be a sequence in $\mathbb{R}$ such that $d(x_n, x_{n+1}) \le \frac{d(x_{n-1},x_n)}{2}$. Show that $x_n$ is a Cauchy sequence. I ...
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Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
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Subsets of a metric space in which Hausdorff semi-distance is symmetric

These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$. ‎‎Hausdorff semi-distance of two subsets ‎$‎A‎, B‎ \subset X$ is defined as below: ‎$‎d(A ...
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Sequential Equivalence Implies Topological Equivalence

Define two metric spaces $(M,d)$ and $(M,\rho)$ to be equivalent, denoted $d\sim p$, to mean that: Topological Definition $\forall x\in M: \forall \epsilon>0 \exists \delta_1>0, \delta_2>0: ...
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if $A$ is open in $M$ and $B$ is open in $N$, then $A \times B$ is open in $M \times N$

where $d((m_1,n_1),(m_2,n_2)) = d_M(m_1,m_2) + d_N(n_1,n_2)$ By some propositions, $A$ is open in $M$ if there exist an open set $K_1$ such that $A = M \cap K_1$ Also, there exist an open set $K_2$ ...
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Prove the following is a metric space…

I need to prove the following is a metric space over the integers: $b \geq 2$. For distinct integers $x, y$. Let $N(x,y)$ be the greatest integer $n$ such that $b^n$ divides $(x - y)$. Let $d(x,y) = ...
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26 views

What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if $x <1/n$}\\ 1 \ \mbox{ if $x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if $x ...
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154 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
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Open set in subspace not open in the entire space example

I am stuck with the following problem: X is a metric space. Suppose that Y is a subspace of X. Give an example that an open set in Y is not open in X. My own approach was this: Suppose U is a subset ...