Let $X=\{1,2,3,4\}$. Then $S=\{\{1,2\},\{2,3,4\},\{2,3\}\}$ is a subbasis for a topology $\tau$. I would like to find the base of $\tau$ generated by $S$.
What is the basis of $ \tau$ generated by S
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general-topology
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0What is the definition of subbasis that you're using? There are two definitions commonly used that are not always equivalent – 2012-10-23
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0@ Bey A subbasis for X, is a collection S of open subsets of X s.t. the collection of all intersections of finite subcollections of S is a base for X. – 2012-10-23
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0I think this definition is not correct: with this definition, a subbasis might not generate the basis for a topology. For example, the collection $S$ you posted above. Based on your definition, the collection $S$ will not generate a basis. My suspicion is that you're accidentally leaving something out of the definition – 2012-10-23
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0@ Bey Let me try again. I'm writing it directly from the text.Let X be a topological space. A subbase for the topology on X,also called a subbase for X,is a collection S of open subsets of X s.t. the collection of all intersections of finite subcollections of S is a base for X. – 2012-10-23
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0@ Bey Maybe the definition includes intersection with itself? – 2012-10-23
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0Ah, yes, taking into account self-intersections will do it. The definition isn't wrong, I just wasn't including those particular intersections (of sets with themselves). Try writing out the basis now – 2012-10-23
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1@Klara: Now that you know that you have to include the members of $S$, you can write up your answer, post it, and (after several hours) accept it; this is encouraged. That will take the question off the Unanswered list and increase your reputation in the process. – 2012-10-23
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0@ Brian, What do you think of my answer. I had a hard time creating the element {1,2,3} and I needed that inside the base. I imagined {1,2,3}=Intersection of {{1,2},{2,3,4}} with {{1,2},{2,3}}. Is this right. – 2012-10-24
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0The set {1,2,3} should not be in the basis. – 2012-10-25
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0@Bey Yes I looked at a definition of checking up a base and I realised I did not need that! – 2012-10-25