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.