Let \(\mathcal{T}\) and \(\mathcal{T}'\) be two topologies on \(X\). If \(\mathcal{T}' \supset \mathcal{T}\), what does connectedness of \(X\) in one topology imply about connectedness in the other?
Solution
Assume \(\mathcal T\subseteq\mathcal T'\), so \(\mathcal T'\) is finer.
If \(X\) is connected in the finer topology \(\mathcal T'\), then it is connected in the coarser topology \(\mathcal T\): any separation by \(\mathcal T\)-open sets would also be a separation by \(\mathcal T'\)-open sets.
Equivalently, if \(X\) is disconnected in the coarser topology, it remains disconnected in every finer topology.
The converse implications fail. For example, \(\mathbb R\) with its standard topology is connected, while the finer lower-limit topology \(\mathbb R_\ell\) is disconnected. Thus connectedness can be destroyed by refinement, but cannot be created by refinement.