Q1. List all possible topologies on the set.
X = {a,b,c}
SOLUTION
. There are 29 distinct topologies on X these are:
1.{X,∅}
2.{X,∅,{a}}
3.{X,∅,{b}}
4.{X,∅,{c}}
5.{X,∅,{a,b}}
6.{X,∅,{a,c}}
7.{X,∅,{b,c}}
8.{X,∅,{a},{b,c}}
9.{X,∅,{a},{a,b}}
10.{X,∅,{a},{a,c}}
11.{X,∅,{b},{a,b}}
12.{X,∅,{b},{b,c}}
13.{X,∅,{b},{a,c}}
14.{X,∅,{c},{a,c}}
15.{X,∅,{c},{b,c}}
16.{X,∅,{c},{a,b}}
17.{X,∅,{a},{b},{a,b}}
18.{X,∅,{a},{c},{a,c}}
19.{X,∅,{b},{c},{b,c}}
20.{X,∅,{a},{a,b},{a,c}}
21.{X,∅,{b},{a,b},{b,c}}
22.{X,∅,{c},{a,c},{b,c}}
23.{X,∅,{a},{b},{b,c},{a,b}
24.{X,∅,{a},{c},{a,b},{a,c}
25.{X,∅,{a},{b},{a,b},{a,c}}
26.{X,∅,{a},{c},{a,c},{b,c}}
27.{X,∅,{b},{c},{a,c},{b,c}}
28.{X,∅,{b},{c},{a,b},{b,c}}
29.{X,∅,{a},{b},{c},{a,b}{a,c},{b,c}}
Q1b. ConsiderS = {{a,b},{a,c}},whereX = {a,b,c}
1
obtain the topology generated by S(i.eτ(S)). Is S a basis?.
SOLUTION
HINT!!. To generate topology from S you will take the possible union of member of S. NOTE THAT!!! ∅ is always there in your Topology that you are generating. Therefore: τ(S) = {X,∅,{a,b},{a,c}}.
And then to know S is basis or not you will use the (3) conditions of topology and test your generated topology (i.e τ(S)). If the (3) conditions satisfies then S is a basis otherwise it’s not a basis.
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