Hyperbolic geometry theorems are abstracted to 

formulate topology theorems for function spaces.

These theorems generalize results which were established 

for complex spaces by

Kiernan, Kiernan -- Kobayashi, Kobayashi, and 

Noguchi. They also provide some new results 

in the framework of complex spaces including 

characterizations of hyperbolically imbedded complex 

subspaces modulo closed complex subspaces in terms of 

the family of holomorphic mappings into such spaces, and 

extension and convergence theorems for this family of 

mappings.