Question

Exercise 31: (General definition of a topology) Let X be a set and O ⊂ P(X), where P(X) := {U ⊂ X}. O is a topology on X iff O satisfies

(i) X∈O and ∅∈O;
(ii) ?i∈I Ui ∈ O where Ui ∈ O for all i ∈ I and I is an arbitrary index set;

(iii) ?i∈J Ui ∈ O where Ui ∈ O for all i ∈ J and J is a finite index set.
In a general topological space O on some X a sequence (xn) ⊂ X converges to x ∈ X iff

for all neighborhoods x ∈ U ∈ O there exists N such that xn ∈ U for all n ≥ N. Show

a) O = ?{a},{b,c},{a,b,c},{∅}? defines a topology on X = {a,b,c}.

b) Write down all possible topologies on X = {a, b, c}.

c) Oc = ?U ⊂ R : R\U isatmostcountableorallofX? defines a topology onX = R. Moreover, show that a sequence (xn) ⊂ R equipped with the topology Occonverges if and only if (xn) is eventually constant, i.e. xn = x for all n ≥ N for some N.

Answer 1