I'm looking for a graduate level book on topology that takes most of its motivation from analysis and applied mathematics. I'm currently in an algebraic topology course but other than basic definitions and intuition I have learned absolutely nothing about algebraic topology!
Now I need to go and relearn most of the topic but it's quite challenging because I find most books on topology unmotivated and uninteresting. Most examples in my class are showing that one sphere is not a different sphere or otherwise use a collection of classic topological objects which I take very little interest in.
I am not trying to diss algebraic topology in any way, but what are some books on topology that stress spaces that are of greater interest to problems in analysis and probability theory?
Ghrist's book "Elementary Applied Topology" looks good, but too cursory for what I'm after. And references on topological data analysis use persistent homology and other topics that are currently above my head.
Some books that I'm aware of but have not read that seem like they may be good are: Lee "Introduction to Topological Manifolds", Dold "Lectures in Algebraic Topology", Rotman "An Introduction to Algebraic Topology", Edelsbrunner "Computational Topology", and Kaczynski "Computational Homology"
If one of the above texts stands out as a good candidate for what I'm interested in, please let me know (It's impossible to read all of them before deciding).
Books that I have read parts of and dislike include: Bredon, Massey, Hatcher, and May.