A topology5 on a finite set X is a subset of TC 2X such that the...

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A topology5 on a finite set X is a subset of TC 2X such that the following holds: 1) If A, BET then An BET and AUBET and 2) {0, X} CT. (a) Prove that 2X is a topology on X and so is {0, X}. (b) Prove that if T is a topology on X and A, B, CET then AUBUC ET too. (c) Find all possible topologies on the set {0, 1}. (d) Find all possible topologies on the set {0, 1, 2}. (e) Prove by induction on #X that there are at least 2 + #X topologies on X. A topology5 on a finite set X is a subset of TC 2X such that the following holds: 1) If A, BET then An BET and AUBET and 2) {0, X} CT. (a) Prove that 2X is a topology on X and so is {0, X}. (b) Prove that if T is a topology on X and A, B, CET then AUBUC ET too. (c) Find all possible topologies on the set {0, 1}. (d) Find all possible topologies on the set {0, 1, 2}. (e) Prove by induction on #X that there are at least 2 + #X topologies on X.