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Suppose S is a sheaf (of abelian groups, say) over a space X. Is there a name for the property: (*) every continuous section over a compact subset K of X extends to a global section? Note that X need not be T_2 etc.

If we replace "compact" by "closed" then we have a "soft" sheaf

If we replace "compact" by "open" we have a "flabby" sheaf

Is there a standard term used in the compact case?

Example: a sheaf over a 0-dimensional space has this property *, but is not necessarily fine or flabby.

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If you replace "compact" by "closed", you obtain a "soft" sheaf, not a "fine" sheaf. ( Fine is a stronger notion: fine $\Rightarrow$ soft.) – Georges Elencwajg Nov 17 '11 at 19:12
Thanks for the correction. Now fixed. – Peter T Nov 18 '11 at 0:34
up vote 2 down vote accepted

The standard term is "c-soft". (See Glen Bredon's book Sheaf theory, for example.)

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