Topological data analyst

I’m currently a Mathematics and Data Science undergraduate who just finished my first year. I’m doing a summer research project in General Topology (my degree focuses heavily on applied math and data science, so we don't have a dedicated topology module, which is why I took on this project).

While my syllabus focuses mostly on statistics and machine learning, I have a very strong interest in pure math. I genuinely enjoy subjects like Real Analysis (which I survived and actually liked!), Calculus, and now General Topology.

Recently, I discovered Topological Data Analysis (TDA). It seems like the perfect bridge between my data science background and my interest in pure math. I was starting to think this could be the perfect field for my future, as my long-term goal is to pursue a PhD and become a researcher/lecturer.

However, while doing some reading last night, I came across a quote in the "Elementary Topology Problem Textbook"(attached picture). This disturbed me quite a bit. I want to ask the experts here:

1. Is this warning mostly about General/Point-Set Topology, or does it apply to TDA and Algebraic Topology as well?
2. Is TDA a worthwhile and active field to study for someone aiming for a career in academic research?
3. If not, what other pure math fields would you recommend for a data science student who wants to bridge the gap between pure math and data?

Any advice would be greatly appreciated!

Author: Wild_Border_7909