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Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a prerequisite for general topology. Someone recommended that I work my way through Spivak's calculus. I'm just wondering if that could be considered a thorough enough intro to analysis that I could start on topology afterward. Also, does Spivak's calculus cover metric spaces?

(I'm still kind of new to PF, so I wasn't sure what prefix to put on my post.)