For my research, I am working with the set $$S = [0,1] \times [0,\delta] \times[0,\delta^2] \times \cdots $$
where $S\subset \mathbb{R}^\infty$. I am using the $\|\cdot\|_2$ norm.
I was hoping to apply Bowers Fixed point Theorem but after some thought I am starting to think this set is not compact. Am I correct that this space is not compact?
My reason from believing that the set is not compact goes as follows:
- Since the set $$S_1 = [0,1]\times [0,1] \times[0,1] \times \cdots$$ is not compact, it has an open cover $C$ with no finite subcover.
- Take each element of the open cover and divide the n-th component by $$\frac{1}{\delta^n}$$ to get $C'$
- Since $C'$ forms an open cover of $S$ (not entirely sure if the sets remain open), and there is no finite subcover, the set is not compact under $\|\cdot\|_2$.
If I am correct that the set is not compact, then are there any fixed point theorems that I may be able to use that many be usefull given my set and norm?
The map you describe will not in general send open sets to open sets since the multiplication by powers of $\delta$ occurs in infinitely many coordinates simultaneously. Your set is compact.