Answer to Question #217054 in Differential Geometry | Topology for Prathibha Rose

Question

Determine all topologies on a set X ={a,b,c}

Answer 1

Let us determine all topologies on a set $X =\{a,b,c\}:$

$\tau_1=\{\emptyset, X\}$

$\tau_2=\{\emptyset,\{a\},\{a,b\}, \{a,c\}, X\}$

$\tau_3=\{\emptyset,\{b\},\{a,b\}, \{b,c\}, X\}$

$\tau_4=\{\emptyset,\{c\},\{b,c\}, \{a,c\}, X\}$

$\tau_5=\{\emptyset,\{a\}, \{b,c\}, X\}$

$\tau_6=\{\emptyset,\{b\}, \{a,c\}, X\}$

$\tau_7=\{\emptyset,\{c\},\{a,b\}, X\}$

$\tau_8=\{\emptyset,\{a\},\{b\}, \{a,b\}, X\}$

$\tau_9=\{\emptyset,\{b\},\{c\}, \{b,c\}, X\}$

$\tau_{10}=\{\emptyset,\{a\},\{c\}, \{a,c\}, X\}$

$\tau_{11}=\{\emptyset,\{a\},\{b\},\{c\}, \{a,b\}, \{a,c\}, \{b,c\}, X\}$

$\tau_{12}=\{\emptyset, \{a,b\}, X\}$

$\tau_{13}=\{\emptyset, \{a,c\}, X\}$

$\tau_{14}=\{\emptyset,\{b,c\}, X\}$

Note that if the topology contains two different 2-element open sets, then by definition it contains also their intersection, that is a singleton.