Advice on topology (undergraduate thesis)



Hello
I'm interested in writing a thesis (undergraduate level)
on point-set or algebraic topology. The only problem is my future
advisor is currently not at my university, he'll be back next month. So
far I got a list of undergraduate thesis topics that he usually gives
advice on, among them are:

1. AANR spaces (AANR= approximative absolute neighbourhood retracts)
2. Duality and homotopy theory
3. Approximative homotopy theory

I took a course on algebraic topology last semester so I'm familiar
with the concept of homotopy (at a basic level) but I have no clue on
what does "Duality and approximative homotopy theory" means and neither
what is the significance of AANR spaces in topology.
I have been reviewing the basics of point-set and algebraic topology,
but I'd really appreciate if someone can explain what does these topics
are about or any introductory books where I can start looking at.
Finally, I'd really appreciate if someone can please provide a list of
thesis topics on topology (besides those I wrote above) to research
about.
I'm particulary interested in point-set topology/algebraic topology but
really I'm not sure on what to write about.

Thanks in advance

.

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