Topology is a canonical example of "mathematics without numbers" (and indeed of "qualitative" rather than "quantitative" mathematics), so for anyone approaching it with an undergrad or lesser background in math, learning it feels a bit like starting with a clean slate. As benrbray mentions in a sibling comment, there aren't any prerequisites in the usual sense, as you won't need to draw on any reserve of mathematical facts built up in learning previous topics while studying it. But learning it does require the ability to assimilate axioms (and to intuit why they make sense as a starting point), and then the ability to reason about those axioms rigorously to draw out the interesting consequences.
That said, a background in some mathematical topics is useful for learning topology. The subject is rooted in a particularly abstract notion of distance, so it's useful to have some experience reasoning with somewhat less abstract notations of distance like metrics, the most familiar of which is the quite numerical Euclidean metric involving the familiar square root of the sum of squares. Familiarity with metric spaces in general (that is, spaces equipped with any axiomatically acceptable notion of a metric distance) is even more useful because it requires a similar sort of axiomatic reasoning as does topology.
Kolmogorov & Fomin's classic text on analysis, the Dover edition of which is under $12 on Amazon [0], has a good (albeit austere) introduction to topology, leading up to it via axiomatic set theory and metric spaces in the book's first chapters. On the one hand, I hesitate to recommend this book, because it was the text for a course that mercilessly exposed the shortcomings in my mathematical education to that time; on the other hand, I recommend it for precisely that reason. Having later assimilated the material, I do consider the book a very good introduction for anyone who either already has the mathematical maturity to do the book properly or for anyone who wants to gain that ability in the way that everybody who has done does: by staring at the same page for hours on end while working everything out on paper, down to the axioms if necessary, until you stop misunderstanding, and then start understanding.
That said, there surely are gentler introductions to the subject if you just want to get a rough idea of it.
That said, a background in some mathematical topics is useful for learning topology. The subject is rooted in a particularly abstract notion of distance, so it's useful to have some experience reasoning with somewhat less abstract notations of distance like metrics, the most familiar of which is the quite numerical Euclidean metric involving the familiar square root of the sum of squares. Familiarity with metric spaces in general (that is, spaces equipped with any axiomatically acceptable notion of a metric distance) is even more useful because it requires a similar sort of axiomatic reasoning as does topology.
Kolmogorov & Fomin's classic text on analysis, the Dover edition of which is under $12 on Amazon [0], has a good (albeit austere) introduction to topology, leading up to it via axiomatic set theory and metric spaces in the book's first chapters. On the one hand, I hesitate to recommend this book, because it was the text for a course that mercilessly exposed the shortcomings in my mathematical education to that time; on the other hand, I recommend it for precisely that reason. Having later assimilated the material, I do consider the book a very good introduction for anyone who either already has the mathematical maturity to do the book properly or for anyone who wants to gain that ability in the way that everybody who has done does: by staring at the same page for hours on end while working everything out on paper, down to the axioms if necessary, until you stop misunderstanding, and then start understanding.
That said, there surely are gentler introductions to the subject if you just want to get a rough idea of it.
[0] https://www.amazon.com/Introductory-Analysis-Dover-Books-Mat...