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I have come across to the following question :

Let $\mathscr{T}_\alpha$ be a family of topologies on $ X$ . Show that there is a unique smallest topology on $X$ containing all the collections $\mathscr{T}_\alpha$ , and a unique largest topology contained in all $\mathscr{T}_\alpha$.

I think that the unique smallest topology equals the union of all the $\mathscr{T}_\alpha's$, and the unique largest topology contained in all $\mathscr{T}_\alpha$ equals the intersection of all the $\mathscr{T}_\alpha's$ .

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    If you show just one of: (i) there is a smallest topology containing all the T (ii) there is a largest topology contained in all the T, then the other statement can be seen as a corollary. It's a little easier to verify (ii). Your idea to use the intersection of the family is the correct one.2011-04-11
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    @Mike: Why are these corollaries of each other? Wouldn't that only be the case if there were a duality where the complement of a topology is a topology?2011-04-11
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    @leopard: A comment I made previously was partly incorrect; I'd misread the question. The correct part of my comment was a hint towards Blah's answer.2011-04-11
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    @joriki: Quite generally if you want to show some poset P which has a greatest and largest element is a complete lattice then you need only check that (i) all subsets of P have a lub, or (ii) all subsets of P have a glb. If, say, (i) holds and you have some subset S of P then consider the set S' of all lower bounds for S. The lub of S' is the glb of S.2011-04-11
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    That should be "some poset P which has a largest element and a smallest element".2011-04-11
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    The union is not a topology, it generates one.2011-04-11
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    @dustin, I don't know if this makes any difference, but this post was asked 3 years ago, wheras the one you mentioned was asked a year ago.2015-03-16
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    @RobertCardona in the older one they were only asking about the smallest whereas this one is about both. So I think this is a more complete question.2015-03-16

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