I am currently reading Munkres' book on topology, in it he defines an open sets as follows: "If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T." Firstly, are the open sets a property of the set X or the topological space (X,T)? Because if the open sets are all the things in the collection T of a particular topological space, different topological spaces on the same set would have different open sets. Making the whole notion of open sets of a set ill defined. Which brings me to my next question: are the open sets the sets in all possible collections T for a set X (in which case defining the open sets of a set makes sense), or are the open sets of a topological space the sets in the collection T of that topological space? In the second case I guess saying the open sets of X would have the fact that the open sets are a property of (X,T) implied. Since also the collection T itself is implied.