Re: Irreducible surjective maps

On 21 Apr., 20:15, Mariano Suárez-Alvarez
<mariano.suarezalva...@xxxxxxxxx> wrote:

Remark that X is assumed to be compact and Hausdorff
in the OP's post.

Let {F_i : i in I} be a decreasing family of
compact subsets in X such that f(F_i) = Z.
Then

  F = intersect_{i in I} F_i

is non-empty.

Let z in Z. For each i in I, there exists
x_i in F_i such that f(x_i) = z. Define a
function phi : I --> X so that phi(i) = x_i.
Consider I ordered so that

  i <= j iff F_i subset F_j.

This makes I a directed set, so that phi
is a net in X. Since X is compact, there
is a convergent subnet, ie, there is a
directed set J and a monotone cofinal map
h : J --> I such that

  psi = phi o h : J --> X

is convergent. Let x in X be its limit
(there is only one, because X is Hausdorff)
Since f is continuous, f(x) = z.
Since the net psi is eventually in every
F_i, which is closed, its limits is in
every F_i, so x in F. Therefore f(F) = Z.

-- m

I liked the proof above, but I think or maybe Im confusing myself.. to
say that psi is eventually in every F_i then shouldnt directed Set I
be downward directed.. i.e. i <= j iff F_j is a subset of F_i? Because
here you want to show that x is a closure point of each of these F_i.
Its just the direction thats confusing me, otherwise Im sure that this
will work.

Sincerely,
Jose Capco
.