I have a question regarding strongly infinite dimensional spaces. Loosely speaking, $X$ is called strongly infinite dimensional if any pair of closed disjoint sets can be separated by a subset $L_i$ and all such subsets have nonempty intersection. If $X$ isn´t strongly infinite dimensional, it is called weakly infinite dimensional.
My questions:
- What does this property intuitively telling us about the space? And why is it called like this, what does it have to do with dimension? It seems to me most examples I can think of, like $\mathbb{R}$ or $l^2$, are strongly infinite dimensional.
- Could you provide examples for strongly dimensional and weakly infinite dimentional spaces?
Thank you for your advice.
For the better context, the definition of a strongly infinite dimensional space is included below.**
