exponentials of perfect maps and local homeomorphisms

[claudio pisani <pisclau@yahoo.it>]

Dear categorists (and topologists),

It is known (Clementino, Hofmann, Tholen, Richter, Niefield...) that perfect
maps p:X->Y are exponentiable in Top/Y, and that the same holds for local
homeomorphisms h:X->Y.

Question: is it true that 1) p=>h is a local homeomorphism) h=>p is a
perfect map?

The conjecture is suggested by the following observations: A) it holds
both for Y = 1 (compact and discrete space) and for subspaces = inclusion
(closed and open parts) B) the analogy between perfect maps and local
homeomorphisms with discrete (op)fibrations (via convergence or other
considerations) and the fact that 1) and 2) above hold in Cat/Y: if p
is a discrete fibration and h is a discrete opfibration then p=>h is
itself a discrete opfibration (and conversely).

Best regards,
Claudio


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