(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. given a set X and a collection of subsets S, prove there exists a smallest topology containing S

2. Prove, on R, the topology containing all intervals of the from [a,b) is a topology finer than the euclidean topology, and that the topologies containing the intervals of the form [a,b) and (a,b], respectively, are not comparable.

3. The attempt at a solution

1. You can say there exists one, the discrete topology. But the smallest one?

2. I tried to write an element (a,b) as the union or intersection of intervals of the form [a,b). You could say [tex](a,b)={\bigcup^{\infty}}_{i=1}[a_{i},b), a_{i}=a+\frac{1}{i}[/tex]

or can you also write:

[tex](a,b) = [\frac{b+a}{2},a)\cup[\frac{b+a}{2},b)[/tex] ?

but then still, why would the latter two be incomparible?

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# Homework Help: Simple topology questions

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