The topologies on {a,b,c,d} up to homeomorphism (in each case, showing all
open sets except  {} and {a,b,c,d} , which of course must be included.)

 1   [{a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, 
	{b, c, d}, {a, c, d}, {a, b, d}, {a, b, c}]  #discrete topology
 2   [{a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {c, d}, {a, b, c}, {a, c, d},
	{b, c, d}]
 3   [{a}, {b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}, {a, b, d}]
 4   [{a}, {b}, {a, b}, {b, c}, {b, d}, {a, b, c}, {a, b, d}, {b, c, d}]
 5   [{a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}]
 6   [{a}, {b}, {a, b}, {a, c}, {b, d}, {a, b, c}, {a, b, d}]
 7   [{a}, {a, b}, {a, c}, {a, d}, {a, b, c}, {a, b, d}, {a, c, d}]
 8   [{a}, {b}, {a, b}, {b, d}, {a, b, c}, {a, b, d}]
 9   [{a}, {b}, {b, c}, {a, b}, {a, b, c}, {b, c, d}]
10   [{a}, {b}, {a, b}, {c, d}, {a, c, d}, {b, c, d}]
11   [{a}, {b}, {a, b}, {a, c}, {a, b, c}]
12   [{a}, {b}, {a, b}, {a, b, c}, {a, b, d}]
13   [{a}, {a, b}, {a, c}, {a, b, c}, {a, b, d}]
14   [{a}, {a, b}, {a, c}, {a, b, c}]
15   [{a}, {b}, {a, b}, {a, b, c}]
16   [{a}, {a, b}, {a, b, c}, {a, b, d}]
17   [{a}, {b}, {a, b}, {a, c, d}]
18   [{a}, {b, c}, {a, b, c}, {b, c, d}]
19   [{a}, {c, d}, {a, b}, {a, c, d}]
20   [{a}, {a, b}, {a, b, c}]
21   [{a}, {b}, {a, b}]
22   [{a, b}, {a, b, c}, {a, b, d}]
23   [{a}, {b, c}, {a, b, c}]
24   [{a}, {a, b}, {a, c, d}]
25   [{a}, {a, b}]
26   [{a, b}, {a, b, c}]
27   [{a}, {a, b, c}]
28   [{a}, {b, c, d}]
29   [{a, b}, {c, d}]
30   [{a}]
31   [{a, b, c}]
32   [{a, b}]
33   []  #coarse topology.

Here are the 68 "minimal" pairs [i,j] such that there is a continuous
injection from W[i] to W[j] (of course all other pairs obtained by
composition could be listed -- there are 346 pairs of these topologies
with one properly containing the other up to homeomorphism). W[1] is
the unique maximal topology and W[33] the unique minimal one, of course.

[1, 2],
[2, 3], [2, 4], [2, 6], [2, 10],
[3, 5], [3, 8], [3, 9],
[4, 7], [4, 8], [4, 9],
[5, 11],
[6, 8], [6, 19],
[7, 13],
[8, 11], [8, 12], [8, 13],
[9, 11], [9, 13], [9, 17], [9, 18],
[10, 17], [10, 18], [10, 19],
[11, 14], [11, 15], [11, 23],
[12, 15], [12, 16],
[13, 14], [13, 16], [13, 24],
[14, 20],
[15, 20], [15, 21],
[16, 20], [16, 22],
[17, 21], [17, 24], [17, 28],
[18, 22], [18, 23], [18, 28],
[19, 23], [19, 24], [19, 29],
[20, 25], [20, 26], [20, 27],
[21, 25],
[22, 26],
[23, 26], [23, 27],
[24, 25], [24, 27],
[25, 30], [25, 32],
[26, 31], [26, 32],
[27, 30], [27, 31],
[28, 30], [28, 31],
[29, 32],
[30, 33],
[31, 33],
[32, 33]