Let X a compact totally disconnected space. We want to show that X is a projective limit of finite spaces. I have an idea, but I can not finish it:
I note I all equivalence relations R on X such that X/R is finite(I is not empty). I was ordered I by the order relation < defined by:
For evry U,V in I; U<V if and only if V included U.
For U <V in I, there f(U,V):X/V-----> X / U defined by f (U, V) ([x] _v) = [X] _U , where [x]_V is the classe of x modulo V.
The system (X/U, f(U,V)) is projectif. I note Y the projective limit of these system.
It is simply to proof that g:X----->Y defined by g(x)=([x]_U), for U in I, is injective and continue.
But I could not show that g is surjective, I tried to show that the immage of g is dense in Y, but I blocked. Somebody help me.
Thank you very much
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