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Consider the set X={a,b}, and the collection £ given by £={ Ø, {a},{b},X }. show that £ is a topology on X.

I know that from the definition of topological space i must consider to show the following three axioms

  1. To show that both the empty set and X belong to £

2.To show that the intersection of any finite sub-collection of sets in £ belong to £

3.To show that the union of any sub-collection of sets in £ belong to £

For the first axiom, it is true that both the empty set and X belong to £, and i have no problem with this.

but my problem lies on second and third axioms,i need to be clear, how can the sub-collections in axioms 1 and 2 be formed?

thanks!

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2 Answers 2

up vote 2 down vote accepted

A sub-collection is a collection of sets in $\mathcal{L}$. For example $\{\{a\},\{b\}\}$ is such a collection. it's intersection is empty and it's union is $X$.

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@ Shamovic;can't i form sub-collection like { {a},{a},{b},{b}}? –  leopard Apr 3 '11 at 13:09
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Since it is a collection, repetition doesn't matter. Order doesn't matter either. –  shamovic Apr 3 '11 at 13:12
    
thanks! –  leopard Apr 3 '11 at 13:14
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{a}$\cap${b} = $\emptyset$ and {a}$\cup${b} = {a,b} = X so both are contained within the topology. This together with the obvious intersections and unions with $\emptyset$ and X show that this is a topology.

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