Since the subject is pretty abstract, you need a certain level of mathematical maturity to grok the concepts. A previous course in abstract algebra or geometry

Since the subject is pretty abstract, you need a certain level of mathematical maturity to grok the concepts. A previous course in abstract algebra or geometry (involving proofs) would be useful. Introductory texts in topology tend to divided into two main subareas - point set topology (the low level axioms of how you define a space and its connectivity properties), and algebraic topology, which focuses on the global mathematical structures of these topological spaces. I found point set topology extremely dry when I first learned it, but it's important to understand those low level basics before moving on to the "cool stuff" in algebraic topology. I recommend "Topology" by Munkres to learn the fundamentals.