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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50 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
4
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0answers
784 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
7
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1answer
387 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
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2answers
183 views

Distance function (without absolute value nor square root)

I'm trying to invent or find a distance function in a two-dimensional space that only makes use of the basic arithmetic operations (+,-,*,/) as I want to use that function in a "programming language" ...
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1answer
85 views

A counterexample on compactness (closed vs complete)

In a metric space $M$: If $A \subset M$ is complete and for each $\epsilon > 0$ there exists a compact $K \subset M$ with $A \subset \{ x \in M : d_M(x, K) \leq \epsilon \}$ then $A$ is compact. ...
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1answer
75 views

Existence of $1/i$-dense subsets

Let $(X, d)$ be a compact metric space and $m$ be a Borel measure on $X$. Assume that $\lbrace A_i\rbrace$ is a nested sequence of subsets: $\dots\subset A_i\subset\dots\subset A_2\subset A_1$ and ...
3
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2answers
187 views

Convergent sequence in product space on $\mathbb{R}^{\omega}$

I am confused about the concept of convergent sequence in product space when learning Munkres's Topology, especially when I am comparing two related exercises of it. The exercise 6 of section 19 ...
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105 views

Describe all the convergent and Cauchy sequences in this metric space

Consider the set of natural numbers $\mathbb N$ with the metric $$d(m,n)=\frac{\left|m-n\right|}{1+\left|m-n\right|}$$ Describe all convergent sequences and all Cauchy sequences in this metric ...
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46 views

$f^{n_i}(x)\to y$ implies $f^{-n_i}(y)\to x$?

Let $(X, d)$ be a compact metric space and $f:X\to X$ be a homeomorphism. If there exists a sequence $n_i$ such that $n_i\to\infty$ as $i\to\infty$ and $x, y\in X$ are such that $f^{n_i}(x)\to y$ as ...
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2answers
95 views

Prove that $|x_1-y_1|+|x_2-y_2|$ is a metric

I have an exercise that states: a) Prove that for $0<p<1$, $d_1(x,y) =(|x_1-y_1|^p+|x_2-y_2|^p)^{1/p}$ is not a metric on $\mathbb{R}^2$ and b) Prove that for $0<p<1$, $d_2(x,y) = ...
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49 views

Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
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1answer
62 views

Compact features

Consider this problem: Let $X$ be a metric space, $U$ be open, $K$ compact and $K\subset U$, show that there exists a $r>0$ such that $B(k,r)\subset U$ $\forall k\in K$ Here $B(k,r)=\{x\in X ...
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0answers
81 views

Is the space $\mathbb N^ \mathbb N$ metrisable? [duplicate]

Given the space $\mathbb N^ \mathbb N$ with the topology generated by basis sets of the form: $$[V,n] = \{x \in \mathbb N^ \mathbb N ; V \text{ is an n prefix of x}\}$$ I can see that this space is ...
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2answers
377 views

Metric space and continuous function

Background: This is an exercise problem from Munkres's Topology (Exercise 3 of Section 20 "The Metric Topology", 2nd edition). It has been posted at this site: Topology induced by metric space. ...
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1answer
169 views

Show that homeomorphism is an equivalence relation in metric spaces

It needs to be shown that homeomorphism is reflexive, symmetric and transitive in all metric spaces. Reflexivity seems to be easy to show, but I'm not sure how to do the rest. Any help?
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1answer
50 views

What is the picture of the basis elements under the metric $d'(x,y) = |x_1 - y_1| + \cdots + |x_n - y_n|$ on $\mathbb{R}^{n}$?

Background: In section 20 "The Metric Topology" of Munkres's Topology, several common metrics are considered on $\mathbb{R}^{n}$. For instance, the euclidean metric $d$ on $\mathbb{R}^{n}$ is defined ...
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1answer
1k views

Is the Cartesian product of two open sets open?

Just a quick question: If you have two sets $A,B \subset \mathbb{R}$ that are open, that is, for every $p \in A$, there exists an $\varepsilon > 0$ such that $B(p;\varepsilon) \subset A$, is the ...
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2answers
882 views

distance between sets in a metric space

I was given this innocent looking homework question. Given two nonempty sets $A,B \subseteq X$ where $(X,d)$ is a metric space. Show that $\mathrm{dist}(A,B) = \inf \{d(x,y) \mid x \in A, ...
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1answer
67 views

Why is the open ball in a discrete space with radius 2 the metric space itself?

I have the definition of open ball as following(dealing with metric space $(M, d)$): Given $x \in M$ and $r > 0$, the set $B_r(x) = \{ y \in M : d(x,y) < r\}$ is called the open ball about $x$ ...
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1answer
74 views

Help using theorem: A function between metric spaces is continuous if and only if for all open sets in the codomain the pre-image of the set is open

As in the title, I am trying to work with the following theorem: $f:M_1 \rightarrow M_2$ is continuous $\iff \forall V \subseteq M_2$ open, $f^{-1}(V)$ is open in $M_1$. As a corollary we showed we ...
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1answer
271 views

Convergence in $L^\infty$ norm and continuous function

Let $\mathcal{C}(T)$ be the set of continuous functions on $T$, which is a metric space under the norm $\left\|f\right\|_{\infty}=\sup_{t\in T}\left|f(t)\right|$. Suppose $\{X_{n}\}$ and $X$ take ...
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1answer
45 views

Let $(X,d)$ be a metric space.Which of the following statements are true?

Let $(X,d)$ be a metric space.Which of the following statements are true? (a)A sequence {$x_n$} converges to $x$ in$X$ iff the sequences {$y_n$} is a cauchy sequence in $X$ , where, for $k\ge1$ , ...
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1answer
79 views

How is the metric space on $[0,1]$ open in itself?

Let $([0,1],d)$ be a metric space. A set is said to be open iff for every element in the set there is some epsilon ball, containing the element, that lies withing the set. $[0,1]$ is said to be open ...
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1answer
60 views

Countable sum of closed boundary sets

I have to prove that for complete metric space and $f_n$ converge pointwisely to $f$ $f^{-1}(a,b)\setminus Int(f^{-1} (a,b)) $ is countable sum of closed, boundary sets. Here is my solution: ...
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1answer
74 views

Proving that a function is a metric

Let $$p(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$$ for $x,y > 0$. Prove that $p$ is a metric for $(0,\infty)$. This question is from Methods of Real Analysis, 2nd edition by Richard ...
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1answer
84 views

Continuity of functions in a metric space.

Let $\mathbb N^{\mathbb N}$ be the set of sequences of natural numbers ($0 \notin \mathbb N$) and let for sequences $a=(n_1,n_2,\ldots)$, $b=(m_1,m_2,...)$ be: $$d(a,b)=\begin{cases} \frac{1}{\min(i: ...
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64 views

An (extended) semimetric on surfaces

Given a smooth surface $S \subseteq \mathbb{R}^3$, like the surface of sphere, we can define the following extended semimetric $d : S^2 \to [0, \infty]$, where $$ d(x,y) = \inf\{\lVert x - p\rVert + ...
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229 views

Which two would be an example of non-equivalent metrics on $\mathbb{R}$. [closed]

Additionally, it's required that neither of them induces the discrete topology.
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Difference between a “topology” and a “space”?

What do we mean when we talk about a topological space or a metric space? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a ...
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64 views

Kullback-Leibler $KL(p,q)\neq KL(q,p)$

I'm doing a course of Artificial Intelligence and in my homework I must to provide a counter example to show that the Kullback-Leibler distance is not a symmetric function of its arguments: $$ ...
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101 views

Pointwise convergence on a complete metric space

Let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
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3answers
317 views

How to prove that a set, by the open cover definition, is not compact

For an exam I have to be able to prove whether certain sets are open, closed or neither and, by extension, (ab)using the Heine-Borel theorem to prove if these sets are compact or not. Because I ...
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1answer
26 views

Proving a straightforward metric space problem

Prove that $|d(a,b) - d(a_{1},b_{1})| \leq d(a,a_{1}) + d(b,b_{1})$ Granted their are two cases to this. I will save one to do independently, but I wanted to see if my proof for the other case is ...
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91 views

Discontinuous function on Q

let $ f_n :X \rightarrow R $ be sequence of continuous function on complete metric space $(X,d)$, which convergence pointwisely to $ f: X\rightarrow R $ mean that $ f_n(x)\rightarrow f(x)$ $ ...
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1answer
46 views

Metric function proof [duplicate]

The time to submit this optional question has passed but I'm curious as to what the answer is. We haven't learned what a metric is yet, which is why its a challenge question. Show that the function ...
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66 views

When is a function space a Fréchet space?

Let $Q$ be a space of indices, and let $(V, |\cdot|)$ be a Banach space of values. Define the function space $X = C(Q,V)$, and equip it with the topology generated by seminorms $\|x\|_D := \sup_{d \in ...
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progression along geodesics

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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Given a metric space $(X,\rho)$, prove that $|\rho(x,z)-\rho(y,u)|\leq{\rho(x,y)+\rho(z,u)}$ for $x, y, z, u\in{X}$.

Obviously it is true, but I'm not sure how to prove it. I'm considering the quadrilateral inequality but so far it has not been helpful. Can anyone give me direction on how to verify ...
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216 views

determinant of the Fubini-Study metric

Is there any easy way to compute the determinant of the Fubini-Study metric, given by: $g_{\alpha\bar{\beta}}=\frac{1}{1+\bar{z}z}\left(\delta_{\alpha\bar{\beta}}-\frac{\bar{z}_\alpha ...
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1answer
102 views

comparison of 3 topologies on C[0,1]

I have a ring of continuous functions from $[0,1]$ to $\Bbb R$. And two norms $C[0,1]\to\Bbb R$. One is supremum of $|f(x)|,$ the other the value of $\int_0^1|f(x)|$. Then I get a Cartesian product of ...
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Embedding finite (discrete) metric spaces to Eulidean space as isometrically as possible

Let $X = \{1, 2, 3, ..., k\}$ with the discrete metric (distance is 1 for every pair of points). How can this be embedded into $\mathbb{R}^n$ (with the usual metric) such that the embedding would be ...
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86 views

I want to prove that the following is a metric space

Let $p \geq 1, a = (a_{1},a_{2}) \in \mathbb{R}^2, b = (b_{1},b_{2}) \in \mathbb{R}^{2}$. Denote $d_{p}(a,b) = (|a_{1} - b_{1}|^{p} + |a_{2} - b_{b}|^p)^{\frac{1}{p}}$. Prove that $(\mathbb{R}^2, ...
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1answer
165 views

X is a metric space. Y is a closed subset of X such that the distance between any two points in Y is at most 1.

I came across this question in an exam I appeared . The question is as follows :- $X$ is a metric space. $Y$ is a closed subset of $X$ such that the distance between any two points in $Y$ is at most ...
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0answers
108 views

Can we have an isometric embedding of this metric space into an Hilbert space?

A metric space (from this Q&A), is defined below. I'd like to know if its possible to have an isometric embedding of this metric space into an hilbert space? As per Schoenberg theorem $-d^2(x,y)$ ...
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1answer
101 views

If the vector space of all real valued continuous functions on the metric space (X,d) is finite dimensional then X is finite set

If $(X,d)$ is a metric space such that $C(X,R)$ is a finite dimensional real vector space, would any one help me to show that $X$ is finite set? $C(X,R)$ denotes the set of all real valued continuous ...
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1answer
97 views

addition and multiplication of functions in function space, continuous?

I have a norm that works in function space of C[0,1]. How do I show that addition and multiplication of functions (C[0,1]xC[0,1]->C[0,1]) are continuous functions?
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3answers
150 views

Can $\le$ be used insted of < in the definition of continuity?

A common definition of a continuous map $T:M_1\to M_2$ is that for every $x\in M_1$ and every $\epsilon>0$ there exists a $\delta >0$ such that for all $y$ in $M_1$ $$d_1(x,y)<\delta \implies ...
3
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2answers
250 views

Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
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1answer
83 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
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1answer
232 views

Complex plane Riemann Sphere topology

Came across the following statement: Define $B_\infty(a;r)$ be the ball in $C_\infty$ with respect to the metric $d_\infty(z_1,z_2) = \frac{2|z_1-z_2|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}$, show that ...