| Axiom | Separate What? | Visual Mnemonic | | :--- | :--- | :--- | | | Two distinct points. | One point is "inside" a set, the other is "outside." They aren't necessarily symmetric. | | $T_1$ (Fréchet) | Two distinct points. | Each point has a neighborhood excluding the other point. Singletons are closed. | | $T_2$ (Hausdorff) | Two distinct points. | They can be "housed" in disjoint neighborhoods. Classic separation. | | $T_3$ (Regular) | A point and a closed set. | A point $x$ and a closed set $A$ (where $x \notin A$) need disjoint houses. | | $T_4$ (Normal) | Two closed sets. | Two disjoint closed sets $A$ and $B$ need disjoint houses. |
Here’s the real gem: Willard’s text has no official solutions because . The only way to “solve” all of them is to develop a personal understanding of topology that is isomorphic to Willard’s own mental model. In category-theoretic terms: