Does there exist a topological space $X$ with the following properties?

- $X$ is connected.
- The set of irreducible components of $X$ is locally finite.
- Not every pair of points in $X$ can be "connected by irreducible components", i.e., there exist points $x,y\in X$ such that there does not exist a finite sequence $(Z_i)_{i=0}^n$ of irreducible components of $X$ with $x\in Z_0$, $y\in Z_n$ and $Z_i\cap Z_{i+1}\neq\emptyset$ for every $i\in\{0,\ldots,n-1\}$.

Note that in such a case, there are infinitely many irreducible components that are not pairwise disjoint.

(I think that such a space must exist: We take an uncountable well-ordered set of irreducible spaces such that each of them meets "the next one" in a single point. However, while unsuccessfully trying to do this rigorously, I got the feeling that some understanding of ordinal numbers might be helpful, which I seemingly do not have; hence the corresponding tag)

This question arose while trying to understand and compare different characterisations of connectedness of topological spaces.

3more comments