Glancing at Appendix A of John Lee's Introduction to Topological Manifolds, I noticed the following list of rules:
I'm familiar with rules like these and have no trouble proving them—that's not my question. Rather, I wonder if there are convenient visuals or general principles that make rules like these easier to remember and even understand. Why do pre-images preserve set-theoretic structure better than images? Why is the structure of unions preserved under images, but not intersections? Again, I'm not looking for a proof of such things because I already know how to prove them. But the proofs are mechanical and don't seem to offer any deep or general insights. Is there a visualization or way of thinking about these rules that makes even the obscure ones, such as (k), quickly comprehensible without the full "mechanical" proof?