Historically speaking, perhaps an analysis course should begin with Fourier's approach to problems regarding the conduction of heat. (For more on this, see e.g. here.)
Of course, many math curricula are organized and re-organized (and re-re-organized) in ways differing radically from their material's original development. For example, Galois Theory is often offered as a sort of "Abstract Algebra II" course, though its creation (e.g. work by Galois himself) did not rely on the group theoretic arguments that you would now likely learn first in an "Abstract Algebra I" class.
As a matter of personal preference, I think it's helpful to start building a topological vocabulary in your first real analysis course (e.g. words such as those mentioned above by Robert Israel). One could argue that learning these definitions in a specific context will be confusing when moving to a more general context (e.g. what compact means in a real analysis course vs. what it will mean in a topology course). Others might argue that having specific examples will help when you go on to study objects of greater generality.
There are a lot of topics that could be covered in an introductory analysis course, and it's quite possible that what is deemed important enough for inclusion is related mostly to the instructor's background (e.g. someone with a background in harmonic analysis might take a more Fourieresque approach in lieu of introducing concepts from point set topology).
I cannot tell you what "should" be done, but I would include a portion on the standard topology in Euclidean space in my own course on analysis, partly because I think the terminology should be seen sooner rather than later, and partly because I find the material particularly interesting and fun to play around with.