Think of the half-open interval $(0,1]$ with the usual open sets (e.g. $(1-\varepsilon,1]$ is an open neighborhood of 1. Then modify the collection of sets considered "open" so that every open neighborhood of 1 contains some set of the form $(1-\varepsilon,1] \cup (0,\varepsilon)$. See if students understand that this modification in which sets are considered open also modifies the way in which the space is connected together.