I am trying to figure out what additional restriction I need to enforce to Cantor's intersection theorem for complete metric spaces to mimic nested interval theorem.
Cantor intersection "theorem": Given any decreasing sequence of non-empty closed nested subsets $\{C_k\}_{k=0}^{\infty}$ with $\color{red}{\text{some additional condition}}$, we have $\displaystyle \bigcap_{k=0}^{\infty} C_k = C$, where $C$ is closed & non-empty.
where $\color{red}{\text{some additional condition}}$ should give me non-emptiness.
Of course, $C$ is always closed. If I enforce, $\text{diameter}(C_n) \downarrow m$, as a counterexample, if we take $\mathbb{R}$ with $d(x,y) = \min \{\vert x- y\vert,1\}$ and $C_k = \left[k,\infty\right)$, we obtain that $C$ is empty, even though $\text{diameter}(C_n) = 1$.
Hence, I want a condition that replaces $\text{diameter}(C_n) \downarrow m$ that ensures that $C$ is non-empty.
However, when $m=0$, we see that $C$ has to be non-empty by completeness.
If we take $C_k$'s to be compact, then the theorem is true in any topological space.
Given that I am on a complete metric space, I want a weaker condition than compactness and it should be of the form closed + "something". If I take "something" = totally bounded, then closed + totally bounded will again give me compactness (since we are on a complete metric space).