In most Descriptive Set Theory books, the rationale for working with the Baire space ($\mathbb{N}^{\mathbb{N}}$) as opposed to the real line ($\mathbb{R}$) is that the connectedness of the latter causes 'technical difficulties'.
My question is, what are these technical difficulties, and why does Descriptive Set Theory (normally?) stick to zero-dimensional Polish spaces?
Thanks in advance.