Let $X$ and $Y$ be topological spaces. The mapping $f:X\to Y$ is continuous if the preimage of the open set is an open set.
"If the topology on $X$ is finer it is "easier" for $f$ to be continuous" ($*$)
Question: As far as I know, the topological space consists only of open sets. For my imagination, in any case , topology being finer or coarser (it's elements are always open) if the preimage of the open set is open then $f$ is always continuous.
I can not imagine how can we say "easier", I mean, how can we compare? Are there any other cases when the topology consists not only of open sets, that make $f$ not "easy" to be continuous? If yes, how can it be topology?
I am really confused now.
I appreciate any help.