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I am curious why it's a problem to define a base using closed sets?

For example, my book uses the definition under "Constructing Topologies from Bases" as specified at http://en.wikibooks.org/wiki/Topology/Bases, as opposed to the "definition" listed on this page. I don't see why closed intervals are a problem for example, the point ${1} \in [0,1], [1,2]$ so in particular $ {1} \in [0,1]\cap[1,2]=[1,1]=\{1\}$
I realize that topologies consist of "open sets" but why can't closed sets be a base for (larger) open sets for a topology.... or more generaly, why can't topologies be constructed using closed sets.

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    I'm confused. Say you have a "base" consisting of closed sets. What are the open sets in the topology you are defining with this "base"?2012-12-29
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    No, my question is why not? You can always fit an open set in a larger closed set. And for example, the set of closed intervals covers $\mathbb{R}$ (which is both closed and open), so I don't see why they are not a base; unless we just require the base to consist of open sets (which is a common definition I've seen).2012-12-29
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    You might want to see [this post](http://math.stackexchange.com/questions/263478/why-there-cant-be-closed-sets-in-topology/263481#263481)2012-12-29
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    Cool... thank you for this. I wasn't aware there was an equivalent definition: 1) The empty set and X are in τ . 2) The intersection of any collection of sets in τ is also in τ . 3) The union of any pair of sets in τ is also in τ .2012-12-29

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