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Exercise 31: (General definition of a topology) Let X be a set and O ? P(X),...

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Exercise 31: (General definition of a topology) Let X be a setand O ? P(X), where P(X) := {U ? X}. O is a topology on X iff Osatisfies

(i) X?O and ??O;
(ii) ?i?I Ui ? O where Ui ? O for all i ? I and I is an arbitraryindex set;

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

for all neighborhoods x ? U ? O there exists N such that xn ? Ufor 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 : RU isatmostcountableorallofX? defines atopology onX = R. Moreover, show that a sequence (xn) ? R equippedwith the topology Occonverges if and only if (xn) is eventuallyconstant, i.e. xn = x for all n ? N for some N.

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