Topology Proceedings 26 No. 1 (2001-2002), pp. 323-341
Controlling extensions of functions and C-embedding
Kaori Yamazaki
We prove that a subspace A of a space X is C-embedded
in X if and only if for every continuous function f:A --> [0, 1] and
disjoint zero-sets Z0, Z1 of X with Zi \cap A=f-1({ i})
(i=0, 1), there exists a continuous extension g:X --> [0, 1] of f
such that Zi=g-1({ i}) (i=0, 1). This extends a result of Frantz
where X is normal and A is closed in X. Applying this result, we
show that some results on controlling extensions of special functions,
which Frantz established on a closed subspace of a normal space, also hold
on a C-embedded subspace of a space. Moreover, we apply the above
result to give new characterization of P\gamma-embedding by extending
suitable collections of functions, and answer a question of Frantz.
