Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $f$, $g$, $h \colon \mathbf{R} \to \mathbf{R}^\omega$ be defined by

$$\begin{align*} f(t)&:=(t,2t,3t,\ldots),\\\\ g(t)&:=(t,t,t,\ldots),\\\\ h(t)&:=\left(t,\frac{t}{2},\frac{t}{3},\ldots\right) \end{align*}$$

for all $t \in \mathbf{R}$. How to determine whether these functions are continuous relative to the uniform topology on $\mathbf{R}^\omega$? We of course assume that $\mathbf{R}$ is given the usual topology.

share|improve this question
2  
What have you tried so far? And is it a homework? –  Thomas E. Feb 7 '13 at 21:47

1 Answer 1

HINTS:

  1. Let $U=\{x\in\Bbb R^\omega:\|x\|<1\}$, where $\|x\|=\sup_{n\in\omega}|x_n|$. $U$ is open in $\Bbb R^\omega$. (Why?) What is $f^{-1}[U]$? Is it open in $\Bbb R$?

  2. Show that for any $s,t\in\Bbb R$, $\|g(s)-g(t)\|=|s-t|$; use this to show that $g$ is continuous. In fact you can use it to show that $g$ is even a homeomorphism.

  3. Show that for any $s,t\in\Bbb R$, $\|h(s)-h(t)\|=|s-t|$.

share|improve this answer
    
please note that here we're using the uniform metric on $\mathbb{R}^\omega$ which is defined as follows: $$\tilde{\rho}(x,y) \colon= \sup\{ \min(\vert x_n - y_n \vert, 1) : n = 1, 2, 3, \ldots\} \forall x \colon= (x_n)_{n \in \mathbb{N}}, y= (x_n)_{n \in \mathbb{N}} \in \mathbb{R}^\omega$$. So do you think you should revise your answer? –  Saaqib Mahmuud Apr 9 at 15:31
    
@Saaqib: It’s the same thing, just a different notation: $\|x-y\|$, as I defined it, is exactly the same as your $\tilde{\rho}(x,y)$. –  Brian M. Scott Apr 9 at 16:45

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.