I strongly second the recommendation of Munkres' Topology. It's basically the gold standard. However, I think don't think it's accessible to junior/senior undergraduates unless they've taken a prior course in analysis. Technically Munkres' Topology has no formal prerequisites, but it's a rough go of it if you haven't had a rigorous exposure to continuity and metric spaces already. The second part of the book also assumes knowledge of elementary abstract algebra.

If the person you're replying to is a junior or senior undergrad in a math major they might be alright (presumably they'll have taken analysis by then). If not, Topology is not what I'd recommend as the first place to learn about metrics and continuous functions (Calculus doesn't cut it for that).