On Fri, 1 Mar 2002, Galchin Vasili wrote:
Hello Cat community,
I have been reading beginning Sheaf theory. The definition of a sheaf varies somewhat from Goldblatt to others.
h A----------->B \ / \ / \ / f \ / g \ / \/ v X
where A, B, and X are topological spaces and X is fixed
Goldblatt: An object is an ordered pair (A, f) where A is a topological space and f is continuous and an etale/local homeomorphism
Other books: The continuity requirement for "f" is dropped.
Why the difference? I don't see that "etaleness" => continuity.
Regards, Bill Halchin
Most people's definition of "local homeomorphism" (including Goldblatt's) implies continuity: if every point of A has an open neighbourhood mapped homeomorphically by f to an open set in X, then the restriction of f to each such open set is continuous, whence f is continuous. Peter Johnstone