Tagged Questions

3
votes
2answers
44 views

Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,…k\}$?

I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for which I've already done a ...
1
vote
1answer
46 views

Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.

I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
3
votes
3answers
39 views

Prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,…,k\}$ is a metric on $\mathbb{R}^k$

I am trying to prove that $d_1\left({x,y}\right) = \max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ is a metric on $\mathbb{R}^k$. So far I know that $\text{dist}\left({x,y}\right) \leq ...
1
vote
0answers
44 views

Determining Complete Metric Spaces

I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$ My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
0
votes
0answers
22 views

Equivalent metrics and inclusion of balls [duplicate]

I need some help with the following proof. I am stuck. My general idea is that if $d_1$ and $d_2$ are equivalent metrics then the balls converge to the same point? However, my understanding of metric ...
0
votes
2answers
56 views

Does this proof make sense and correct — is it written well enough?

I'm working on a tutorial question. The question asks whether the following claim is true or false, if it is true: one is supposed to provide a proof or counter-example otherwise if it's false. Let ...
1
vote
1answer
141 views

Show $\mathbb R^n$ is complete.

Show $\mathbb R^n$ is complete. At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far: ...
0
votes
0answers
77 views

What's wrong with my proof that a continuous function is uniformly continuous?

I was trying to prove that any continuous function from a compact metric space to any other metric space is uniformly continuous. I proved it as follows: Let $f\colon X\to Y$ be continuous and ...
6
votes
3answers
411 views

'Every open set in $\mathbb{R}$ is the union of disjoint open intervals.' How do you prove this without indexing intervals with $\mathbb{Q}$?

In my book's exercises section I am asked to prove that every bounded, open set in $\mathbb{R}$ is the union of disjoin open intervals. Looking around the internet I have found many strategies that ...
0
votes
2answers
117 views

Metric spaces question

\begin{equation} P = \{f \in\ C ( \mathbb{R}, \mathbb{R}) \mid f(x+2 \pi ) = f(x)\} \end{equation} be the set of $2\pi$-periodic function. 1) Show that $P$ is a subspace of $C( \mathbb{R}, ...
0
votes
0answers
136 views

Isometries of metric spaces questions

A function $f$ from a metric space $(E, d)$ onto a metric space $(Y,\tilde d)$ is called an isometry if $$ \forall x, y \in E : \tilde d(f(x),f(y)) = d(x,y). $$ Show that the function $f:(0,1] ...
4
votes
4answers
101 views

Show that the set given is closed

A question that I encountered which looks different than a normal open/closed sets proofs: Let $(E, d)$ be a metric space, let $f : E\to R$ be continuous and $a$ element of $R$. Show that the set ...
3
votes
4answers
165 views

Show that the set is closed

Let $(E, d)$ be a metric space, $x$ element of $E$. Show that the set \begin{equation} A = \{y \in\ E : d(x, y) \geq 5 \} \end{equation} is closed. Generally, how would you go about this? I have an ...
1
vote
1answer
282 views

Topology: Proving a space is connected

I'm attempting to prove that a space is connected and compact. I have a continuous function $f:X \rightarrow S^{1}$. $X$ is metrizable and locally connected. $f$ is non-constant, surjective and ...
3
votes
1answer
279 views

On two different proofs

This question is mainly to understand the meaning of my professor's correction to a proof of a theorem I gave during an oral examination. The question was to show that $\text{diam}A = ...