Activities for part 2

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

• 0 · Distribute compatible weightsA measure gives a weight to every box at every stage. Each parent weight must equal the sum of the child weights inside it.
• 1 · Compare ordinary and p-adic distanceAdd 1, then p, then p squared, and continue with higher powers. The totals grow under ordinary distance but settle toward a limit under p-adic distance.
• 2 · Integrate at a coarse or fine stageA continuous integer-valued function is constant on the boxes of some finite stage. We can repeat its values on finer boxes without changing the function.
• 3 · Build a function from simple piecesFor each box, an indicator function is one on that box and zero on all the others. Integer multiples of these simple functions can build any integer-valued function on a finite stage.
• 4 · See which coordinates a map can readA product may have a nonzero integer in every coordinate. Specker's theorem says that an additive map from a countable product to the integers can read only finitely many of them.
• 5 · Check the unique-extension ruleA condensed group is solid when every assignment on probe points extends in exactly one way to all compatible integer measures.
• 6 · Compare divisible groupsA divisible group allows division by every positive integer. The additive real numbers have this property, but the integers do not.
• 7 · Read a completed tensor productThe completed tensor product combines two solid groups and keeps completion directions that are compatible with both sides.
• 8 · Recover homology from solidificationDerived solidification of the free condensed group on a space recovers its integral homology. The bars separate free classes from classes of finite order.

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