Question

Q/State all possible topologies on set X ={1,2,3, 4}.

Answer 1

All possible topologies on the set X=$\{1,2,3,4\}$ is all possible collection of subsets which contain finite intersection and arbitrary union of the power set of $\{1,2,3,4\}.$ Its power set has $2^4=16$ elements. In each topology the empty set and the entire set is there. Rest we must check options.

We just note

$\tau =\{\emptyset,\{1\}, \{1,2,3,4\}\}$ and $\tau =\{\emptyset,\{a\}, \{1,2,3,4\}\}$ are isomorphic topologies where a is 1,2,3,or 4.So we just write once their equivalent class. The classes are too many.

$\tau=\{\empty , \{1\},\{1,2,3,4\}\}$ ,

$\tau=\{\empty , \{1,2,3\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\},\{1,2,3\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\}, \{1,2,3\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\}, \{1\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\}, \{1\}, \{2\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{4\}, \{1,2,3\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{1,2,3\}, \{1,4\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{1,2,3\}, \{4\}\{1,4\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{2,3\}\{1,2,3\}, \{1,4\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\}, \{1,2,3\}, \{1,2,4\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{1,2,3\}, \{1,4\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\}, \{3\}, \{1,2,3\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1,2\}, \{3\},\{1,2,3\}, \{1,2,4\},\{1,2,3,4\}\}$

$\tau=\{\empty , \{2,3\}, \{1,4\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{1,2\},\{1,2,3\}, \{1,2,4\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{1,2\}, \{1,2,3\}, \{1,3\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{2\}, \{1,2\}, \{1,2,3\}, \{1,3\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{1,2,3\}, \{1,2\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{2\},\{1,2,3\}, \{1,2\}\{1,2,3,4\}\}$

$\tau=\{\empty , \{1\}, \{3\},\{1,2,3\}, \{1,3\},\{1,2,4\},\{1,2\},\{1,2,3,4\}\}$

$\tau=\{\emptyset,\{1\},\{1,2\},\{1,3\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{1,3\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{2,3\},\{1,2,3\},\{1,4\},\{1,2,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{1,2\},\{1,3\},\{1,2,3\},\{1,4\},\{12,2,4\},\{1,3,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\}.\{1,3\},\{1,2,3\},\{1,4\},\{1,2,4\},\{1,3,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{3\},\{1,3\},\{2,3\},\{1,2,3\},\{1,2,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{3\},\{1,3\},\{2,3\},\{1,2,3\},\{1,4\},\{1,2,4\},\{1,3,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{3\},\{1,3\},\{2,3\},\{1,2,3\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1\},\{2\},\{1,2\},\{3\},\{1,3\},\{2,3\},\{1,2,3\}, \{4\},\{1,4\},\{2,4\},\\ \{1,2,4\},\{3,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\}\\ \tau=\{\emptyset,\{1,2,3,4\}\}$