The solution to the problem of proving that the continuous image of a connected space is connected (a classic exercise in Ryszard Engelking’s General Topology
Engelking’s text is unique because its exercises are not merely for drill—they often contain that act as sketches for proofs. The problems are frequently organized into series that span multiple chapters, covering advanced topics such as: Linearly ordered spaces Cardinal functions and their relationships Spaces of closed subsets (hyperspaces) Semicontinuous functions and set-valued mappings Key Problem Areas and Solving Strategies
The ultimate is the ability to construct solutions de novo . Here is a 5-step framework for any Engelking problem:
This is simple, yet half the online "solutions" forget to state that perfect normality includes normality as a hypothesis, leading to a circular argument.