As we will mention repeatedly (and something the reader should always keep in the back of their mind to ground themselves when things seem confusing), topology generalises the notions of distance or even more precisely, it generalises the notions of "closeness". One can see why topology is often considered the loosest structure one can impose on a set (other structures being groups, rings, etc. which provide very strong relations between elements).

The reader may ask why we wish to generalise these notions or why generalising them is useful. There may be many motivations but we provide one here. Consider, for example, differentiation, which allows us to measure how a function behaves close to a given point. Doing so makes many problems easier since we can get a nice approximation of the function and it also provides us with useful information about the function (e.g. whether it is increasing or decreasing, existence of extrema, etc.). In order to similarly gain information in more general contexts, we at least need some notion of "closeness".