I found that the comment box underneath andrews response wasnt large enough for what i had to say. I think that before i continue in my answer i should mention that i study homotopy theory, and maybe that is why i dont really care about motivating the "original" definition of a topological space. In homotopy theory, and perhaps any geometric flavor of topology, we work with things that have the homotopy type of a CW-complex, these may be much easier to motivate.

I think that the best way to motivate the definition in terms of open sets is historical (i think this is often the case, when you look at what people were thinking about or the problems they were trying to solve or overcome the definition might become clearer). When people started writing down the definition of what a topological space was there was a strong penchant for axioms and set theory. This is the flavor of the definition in terms of open sets. The definition that we have in terms of open sets was gotten after a bit of hard work with bad definitions. There was a lot of change in the culture of mathematics at the turn of the century and a lot of things had to be reworked and made rigorous. Perhaps i have the facts wrong, but it makes some sense this way even if i am mistaken.

One of my instructors frequently answers questions by saying things like we dont care about that or that is a bad question, which i feel is a legitimate response. The point is that there is a lot of mathematics to be done, a lot of really beautiful important mathematics. You can't really do all of it in a lifetime, so it is probably good to accept some simplifying assumptions like your ring is Noetherian or your space has the homotopy type of a CW-complex. The objects you are ignoring are not that natural to begin with and the things you are looking at are really much more important. In the end the questions we don't answer about the topologists sine curve won't really matter (...I think?) How could you hope to answer a question about some pathological special example with a tool that is meant to capture intuition?

since i dont know how to save this answer as a draft i will just have to settle for coming back to edit it later