Problem
Give an example of a topological space $(X, \mathcal{T})$ for whose hyperconnected components $\mathcal{C} \subset \mathcal{P}(X)$ it holds that $\mathcal{C}|D$ is not $(\mathcal{T}|D)$-locally finite for some $D \in \mathcal{C}$. Here $\mathcal{T}|D$ denotes the subspace topology on $D$, and $\mathcal{C}|D = \{C \cap D : C \in \mathcal{C}\}$ is the restriction of a cover to $D$.
Background
- The excluded point topology on an infinite set $X$ and point $p \in X$ provides an example where hypercomponents are not $\mathcal{T}$-locally finite. However, in that example $\mathcal{C}|D = \{\{p\}\}$ for any $D \in \mathcal{D}$, which is $(\mathcal{T}|D)$-locally finite, and so is not a valid example for the current problem.
- Hypercomponents may intersect, and there may be infinity of them, as shown in the above example.
- Hypercomponents are closed.
- An algebraic set has a finite number of hypercomponents (under the Zariski topology), so do not provide an example.
- Ultimately, I'm trying to solve this related question. Showing that there is no such space as in the current problem would resolve that question. Providing an example of such a space would provide some intuition for that question.
Edit: After no answers, I have asked this same question at Mathoverflow.