I am new to both subject areas and am currently choosing modules to take next year. I'm pretty certain I want to take topology but metric spaces is currently not on my list. Would it be recommended to take metric spaces as a kind of pre-requisite to topology or am I fine taking just topology. Any information on the role metric spaces play within topology or how integral it is in understanding topology?
Metric spaces are particularly nice topological spaces. Many topological spaces found “in the wild” are metric spaces. Almost every topological space I work with in my particular field is a metric space.
To answer your questions:
Understanding metric spaces is not a prerequisite to understanding topology, although you do learn many topological ideas through studying metric spaces. It can be a good jumping off point to general topology.
Conversely, you do not need to understand metric spaces in order have a basic grasp on topology. Most likely, you will learn what a metric space is in your topology class though you won’t go as in depth as your metric spaces class might.