I have what I hope to be just a simple notation/definition question I can't seem to find an answer to. I'm not going to post my hw question, just a piece of it so I can figure out what the question is actually asking. I have a function i:A --> X I also have a continuous function g: A --> B. Then I am asked to prove a property about the "induced map" f: B --> B Ug X I am just having trouble understanding exactly what this "induced map" is. There are no defs for it in my book and online I only see induced map as induced homeomorphisms. So my question: is this a quotient map? Is B Ug X just a quotient space? My map i is not necessarily an inclusion, so A and X could be two separate spaces, so I"m assuming this is a quotient space because an a in A could map to X under i but could also map to B under g. I guess I'm just confused about this function. Does i even factor in to this map? Also when I think of the composition f(i(A)) what on earth this would be like.