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

Question #217054

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


1
Expert's answer
2021-07-16T02:24:01-0400

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


τ1={,X}\tau_1=\{\emptyset, X\}


τ2={,{a},{a,b},{a,c},X}\tau_2=\{\emptyset,\{a\},\{a,b\}, \{a,c\}, X\}


τ3={,{b},{a,b},{b,c},X}\tau_3=\{\emptyset,\{b\},\{a,b\}, \{b,c\}, X\}


τ4={,{c},{b,c},{a,c},X}\tau_4=\{\emptyset,\{c\},\{b,c\}, \{a,c\}, X\}


τ5={,{a},{b,c},X}\tau_5=\{\emptyset,\{a\}, \{b,c\}, X\}


τ6={,{b},{a,c},X}\tau_6=\{\emptyset,\{b\}, \{a,c\}, X\}


τ7={,{c},{a,b},X}\tau_7=\{\emptyset,\{c\},\{a,b\}, X\}


τ8={,{a},{b},{a,b},X}\tau_8=\{\emptyset,\{a\},\{b\}, \{a,b\}, X\}


τ9={,{b},{c},{b,c},X}\tau_9=\{\emptyset,\{b\},\{c\}, \{b,c\}, X\}


τ10={,{a},{c},{a,c},X}\tau_{10}=\{\emptyset,\{a\},\{c\}, \{a,c\}, X\}


τ11={,{a},{b},{c},{a,b},{a,c},{b,c},X}\tau_{11}=\{\emptyset,\{a\},\{b\},\{c\}, \{a,b\}, \{a,c\}, \{b,c\}, X\}


τ12={,{a,b},X}\tau_{12}=\{\emptyset, \{a,b\}, X\}


τ13={,{a,c},X}\tau_{13}=\{\emptyset, \{a,c\}, X\}


τ14={,{b,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.


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