All Questions
Tagged with metric-spaces solution-verification
964
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Prove that the usual metric and other metric induce the same topology
I am working on A course on Borel sets, by S.M. Srivastava. There is this problem I am working on that states the following:
Show that both the metrics $d_1$ and $d_2$ on $\mathbb{R^n}$ defined in 2.1....
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Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Let $(X,d)$ a metric space. Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Here's my proof so far, please check it.
Proof.
Assume that $K \subseteq ...
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Basis for a topology in $M_{n \times n}(\mathbb{R})$
I'm starting my studies in topology (on my own) and sometimes I have a hard time seeing how to use the concepts. I need help with this problem:
Let $X$ be the set of all ($n \times x$) matrices of ...
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To prove that the diameter of a solid triangle is the length of the largest side
This is a preliminary in the proof of Cauchy-Goursat Theorem for Triangles in complex analysis,
The solid triangle enclosed by $a,b,c$ in the complex plane is defined as $$\Delta(a,b,c):=\{t_1a+t_2b+...
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A separable normed space that is continuously embedded in a non-separable normed space implies that this embedding isn't dense.
As a preliminary I introduce the definition of denseness I am using:
Definition (dense subsets of metric spaces). Suppose $(M,d)$ is a metric space. A subset $S \subset M$ is called dense in $M$ if ...
- 1,141
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1
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77
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Convergence of a sequence w.r.t French Railway Metric
So I'm given the following version of the French Railway Metric:
$$
d^{'}_{f}(z_1, z_2) =
\begin{cases}
|z_1 - z_2| & \text{if } \exists \lambda \in \mathbb{R} \text{ such that } z_1 = \lambda ...
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Proving $d_ud_vf = d_vd_uf$ if $f$ is $C^2$ for any vectors $u,v$ using Clairauit's Theorem:
I was wondering if a proof using Clairaut's theorem works. Here is my attempt; it looks right to me but I am not comfortable with differential calculus in Euclidean space at all still.
Let $f: \mathbb{...
- 1,774
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Every subset of discrete space is open and closed.
Had a look at the solutions and their answer is different. Was wondering if my attempt works and would love some feedback? Thank you in advance!
Question:
Let $X$ be a non-empty set equipped with ...
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1
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179
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Proof that if $K$ compact then $ \text{diam} (K) < \infty $
I have the following question but without any correction so I will really appreciate a feedback on my answers.
Question:
Let be $K$ a non-empty compact metric space. We define $ \text{diam} (K) = \sup ...
user1265502
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An inductive proof for the Heine-Borel theorem in $\mathbb{R}^n$
The Heine-Borel theorem for $\mathbb{R}^n$ states that a subset $K\subset\mathbb{R}^n$ is closed and bounded if and only if it is compact. (A set is said to be compact if every open cover of the set ...
- 3,044
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Chapter 2.3 Exercise 14 - Metric Spaces: A Companion to Analysis
I am trying to prove one of the questions in Metric Spaces: A Companion to Analysis, Chapter 2.3, Exercise 14, but am trying to make sure I am understanding this correctly.
Let $X$ be a metric space ...
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Question about proper notation for an exercise concerning functor for metric space in Arbib and Manes
The following question is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes
$\color{Green}{Background:}$
$\textbf{(1a)}$ $\textbf{Definition:}$ A ...
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The closure of the closure
I would like general feedback on my solution to this exercise, as well as any ideas for a more direct approach.
Exercise
Let $(X,d)$ be a metric space. Denote the closure of any set $A\subseteq X$ by $...
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If $A, B \subseteq X$ and $A \cap B \neq \emptyset$, then $\text{diam}(A \cup B) \leq \text{diam}(A) + \text{diam}(B)$
I am starting to learn about Metric Spaces and I am trying to have a look at some exercises and came across this question:
Let $(X, d)$ be a metric space, and let $A, B \subseteq X$. If $A \cap B \...
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Showing that a visual ball is contained in the shadow of a ball.
Let $X$ be a Hadamard manifold of negative curvature $K_X \le -1$. With respect to the visual metric (with respect to the origin $o$) on the boundary at infinity $\partial X$, consider a visual ball $\...
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