Right now my friend and I are going through Munkres Topology and we meet up to work on the challenge problems. I am wondering if anyone knows of a good resource or if someone could post some difficult questions. Right now we are mostly focusing on stuff before the quotient topology in Munkres.
Any help much appreciated.
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I recall the exercises in Spanier's Algebraic topology being quite challenging. Each chapter contains on average 40 exercises, which are grouped by particular concepts (e.g. "Local systems", "The index of a manifold").
I find that having such small selections of exercises around one particular concept can be very useful. The challenge in an exercise primarily lies in coming to terms with one particular concept, and different exercises allow one to see this concept from different angles. Solving groups of exercises allows for deep absorption of the relevant concept and is also time efficient (as one always needs time to "break in" to another concept, so switching between unrelated exercises always costs time).
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Read through the classic Counterexamples in Topology and fill in the relevant details as you go along. Many interesting examples of spaces are given, but often without the detailed proofs that they have the properties that the book claims that they have (since that would bloat the book and make it less useful as a reference).
Try this book: Fundamentals of General Topology, Alexander Arhangel’skii.
As the author has written in the preface that
The present book bearing the title, Fundamental of General Topology: Problems and Exercises, is addressed to the mathematicians interested in, and students of topology...
The book is highly suitable for directing the reader toward rigorous research, giving him at the same time a systematic training to actively analyze other mathematical texts...
So I think this book serves your need.