Example of chainable space

Question

A ‎chain in a space ‎$‎X‎$‎ is a finite family of open sets ‎$‎\mathcal C = \{C_1,C_2,\ldots,C_n\}‎$‎ such that ‎$‎C_i\cap C_j \not=‎\emptyset‎‎$‎ if and only if ‎$|i -‎ j|\leq1‎$‎. Space ‎$‎X‎$‎ is chainable provided that every open cover of ‎$‎X‎$‎ can be refined by a chain covering ‎$‎X‎$‎.

Can anybody supply an example of a chainable space.

Answer 1

Clearly such a space must be compact. Let $\langle X,\le\rangle$ be a linear order, give $X$ the associated order topology, and suppose that the resulting LOTS (linearly ordered topological space) is compact. It’s a nice little exercise, not too hard, to show that $X$ is chainable. (First observe that any open cover of $X$ has a finite refinement by open intervals. This refinement has an irreducible subcover, meaning that removing any member of the subcover leaves some point of $X$ uncovered. Now verify that the subcover must be a chain.)

Added: I overlooked the requirement that adjacent members of the chain must intersect. That need not be the case if $X$ is not connected. In fact, if $X$ is not connected, it is the union of disjoint, non-empty open sets $U$ and $V$ such that $x<y$ whenever $x\in U$ and $y\in V$, and the open cover $\{U,V\}$ has no refining chain. Fortunately, it is sufficient to assume that $X$ is connected, as then adjacent open intervals must in fact overlap.

This isn’t the only way to produce chainable spaces, but it’s a nice general way that produces a lot of examples.

It may also be worth looking for compact spaces that aren’t chainable, to get a better idea of just what chainability does for you. One rather easy example is $S^1$, the circle.