continuity of a function; uniform topology on R^\omega

**abhishekkgp** · January 4th 2012, 02:02 AM

Consider $\mathbb{R}$ in the standard topology and $\mathbb{R}^{\omega}$ in the uniform topology.
Consider the function $g:\mathbb{R} \Rightarrow \mathbb{R}^{\omega}$ given by $g(t)=(t,t,t, \ldots)$ .
QUESTION: Is $g$ continuous??

MY ATTEMPT:

Let $\bold{x} \in \mathbb{R}^{\omega}$ where $\bold{x}=(x_1,x_2,\ldots)$ .

Let $1> \epsilon >0$ .

To see whether $g^{-1}(B_{\overline{\rho}}(\bold{x},\epsilon))$ is open in $\mathbb{R}$ or not.

I have worked out that $B_{\overline{\rho}}(\bold{x},\epsilon)=\bigcup_{ \delta < \epsilon}U(\bold{x},\delta)$ , where

$U(\bold{x},\delta)=(x_1-\delta ,x_1+\delta )\times (x_2-\delta ,x_2+\delta )\times \ldots$

Using this i proved that $g^{-1}(B_{\overline{\rho}}(\bold{x},\epsilon))=\bigcup _{ \delta < \epsilon}\left(\bigcap_{i \in \mathbb{Z}^+}(x_i- \delta, x_i + \delta) \right)$

I am not able to see whether $\bigcup_{ \delta < \epsilon}\left(\bigcap_{i \in \mathbb{Z}^+}(x_i- \delta, x_i + \delta) \right)$ is open in $\mathbb{R}$ or not. Can someone please help me on this??

**abhishekkgp** · January 4th 2012, 04:15 AM

Okay i think i've got it. 'g' is continuous.
Anyways.. thanks to anyone who cared to have a look.

**Amer** · January 4th 2012, 06:03 AM

why you did not try to prove that g is continuous at every point x in R

g is continuous at x ( x in the domain sure ) if for every open set "v" containing g(x) we can find an open set "u" in
R such x $\in$ u and g(u) $\subset$ v

how is the open sets in the uniform topology ?

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