Problem 2-2 in Lee's Introduction to Topological Manifolds reads:
Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that every topology on $X$ is homeomorphic to exactly one on your list.
Below is my attempted solution. Is there a more efficient way to solve a problem like this one? My method felt a bit tedious.
The topologies can be partitioned into classes according to how many 1- and 2-element sets they contain. Let us write $(m, n) \in \{0, \dotsc, 3\}^2$ to represent each such class, with $m$ being the number of 1-element sets (singletons) and $n$ the number of 2-element sets in the topology, and go through each of them. Note that homeomorphisms preserve class membership.
(0,0) This class contains only the trivial topology $\{X, \varnothing\}$.
(1,0) If we add a single singleton to the trivial topology we obtain a topology homeomorphic to $\{X, \varnothing, \{1\}\}$.
(2,0) This class is empty, since if we have two singletons we must also have the 2-element set given by their union.
(3,0) Similarly empty.
(0,1) Adding a 2-element set to the trivial topology we obtain a topology homeomorphic to $\{X, \varnothing, \{1, 2\}\}$.
(1,1) There are two distinct ways we can add a singleton and a 2-element set to the trivial topology; either they overlap or they do not. Hence we either get a topology homeomorphic to $\{X, \varnothing, \{1\}, \{1, 2\}\}$, or one homeomorphic to $\{X, \varnothing, \{1\}, \{2, 3\}\}$. Note that these are not homeomorphic to each other. (A homeomorphism would have to map $\{1\}$ to $\{1\}$, but simultaneously map $\{1,2\}$ to $\{2,3\}$, which is impossible.)
(2,1) All topologies in this class are hoemorphic to $\{X, \varnothing, \{1\}, \{2\}, \{1, 2\}\}$, since a topology with two singletons must also contain their union.
(3,1) All three singletons together generate the discrete topology, so this class is empty.
(0,2) If we have two 2-element sets they must overlap on a single element, thus giving us also a singleton via closure under finite intersection. Hence this class is empty.
(1,2) By the logic above, the topologies in this class are all homeomorphic to $\{X, \varnothing, \{1\}, \{1, 2\}, \{1, 3\}\}$.
(2,2) These topologies must have a (1,2)-topology as a subset, by the logic above. Hence they are homeomorphic to $\{X, \varnothing, \{1\}, \{2\}, \{1, 2\}, \{1, 3\}\}$.
(3,2) Again, only the discrete topology contains all three singletons.
(0,3) With all three 2-element sets we get all the singletons via intersections, hence this class is empty.
(1,3) Similarly empty.
(2,3) Similarly empty.
(3,3) Here we have only the discrete topology, $\mathcal P(X)$.
Hence every topology on $X$ is homeomorphic to exactly one of the following topologies: $$\{X, \varnothing\},$$ $$\{X, \varnothing, \{1\}\},$$ $$\{X, \varnothing, \{1, 2\}\},$$ $$\{X, \varnothing, \{1\}, \{1, 2\}\},$$ $$\{X, \varnothing, \{1\}, \{2, 3\}\},$$ $$\{X, \varnothing, \{1\}, \{2\}, \{1, 2\}\},$$ $$\{X, \varnothing, \{1\}, \{1, 2\}, \{1, 3\}\},$$ $$\{X, \varnothing, \{1\}, \{2\}, \{1, 2\}, \{1, 3\}\},$$ $$\mathcal P(X).$$
