Activities for part 1

Choose a chapter activity below. Each page turns one mathematical idea into a finite example that you can change.

• 0 · Build a probe from finite stagesA probe begins with one box. Each new stage divides the boxes into smaller ones. A point in the finished probe is a path that makes one compatible choice at every stage.
• 1 · Compare two ideas of closenessThe two rows contain the same real numbers. In the top row, every point stands alone. In the bottom row, the numbers have their usual distance, so nearby values remain close.
• 2 · Compare three kinds of probesThese diagrams show finite stages of a halving probe, an approaching sequence, and a base-p probe. Every box has a precise parent in the stage above it.
• 3 · Check continuity with a sequenceThe sequence points approach a limit. A continuous map must send those points toward the value assigned to the limit point.
• 4 · Use the cut and glue rulesA condensed set treats separate pieces independently. It also joins answers on a cover when they agree wherever the cover repeats the same point.
• 5 · Find a quotient that points missA probe can support a continuous real-valued function that is not locally constant. Such a function gives information that the one-point probe cannot detect.
• 6 · Try to choose a continuous sectionA cover gives possible points above each point of a probe. A section chooses one point above each point, and those choices must vary continuously.
• 7 · Recover a familiar spaceFirst record every continuous map from a profinite probe into a space. Then use those probe maps to recover a topology on its points.
• 8 · Keep track of windingA closed path around a circle has an integer winding number. A continuous deformation may change the path's shape, but it cannot change that number without breaking the path.

Every activity is self-contained and uses no network connection or external library. You can therefore save and run the pages offline.