(unknown)

[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

2) 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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