7 · Recover a familiar space

First record every continuous map from a profinite probe into a space. Then use those probe maps to recover a topology on its points.

Try this. Send the circle, interval, and finite set through both steps. Each one returns with its original topology. The final example stops because it has a point that is not closed.

Proposition 1.7 proves this recovery for compactly generated spaces, and Theorem 2.16 gives the compact Hausdorff case. Warning 2.14 explains why the rejected example lies outside the assumptions.