I am a second year undergraduate college student interested in applied math program. I hear a lot that general topology(e.g. the first half of Munkres' book Topology) is very useful, but is it really helpful for people like me who are into "applied" math, rather than pure math??
For example in data analysis one could use topological approach. And they actually do. Look the cloud of data (very big data) may have some topological characteristics: cycles, wholes, components, and so on.
Traditional statistical tools are not robust enough to deal with certain high dimensional and noisy data. We need additional methods that we can use to modify and preprocess the data to a form which is more suitable for statistical tools. One very promising direction is to use topology. It was Gunnar Carlsson and Herbert Edelsbrunner who first realize the potential of topological tools to obtain new qualitative information about large dimensional data sets. For example the topological data analysis was essential to identify a subgroup of breast cancers with a unique mutational profile and excellent survival, information invisible to classical methods.