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During my last fall semester, I was taking a course in point set in topology, and thought algebraic topology would be a great course to take in the spring semester. However, in my topology course we only got through the metric topology(3rd chapter). In result, I feel that I am not prepared to do well in algebraic topology, since I did not learn much from point set topology. I have two and a half weeks before the next semester starts, and I wanted to ask what should I read heavily on to be prepared for the course?

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An algebraic topology course has a very different focus from a point-set topology course and shouldn't really be viewed as a "continuation." I don't think you have to worry too much about the details of a point-set topology course you may not have reached. Perhaps it's better to review the topological basics and make sure you're comfortable with them.

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I think there are some wonderful free notes on the basics and prerequisites of point-set topology for Algebraic Topology written by Hatcher: https://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf

They are short, go straight to the point and there are plenty of amazing and clarifying pictures.

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