PERFECT MAPS IN COMPACT ( COUNTABLY COMPACT ) SPACES

In this paper, among other results, characterizations of perfect maps in compact Hausdorff(Frchet, countably compact,Hausdorff) spaces are obtained

continuous or onto unless mentioned explicitly,cl(A) will denote the closure of the subset A in the space X.A map f:X--.Y is said to be countably compact (compact[I] or proper in the terminology of Liden [2]) if inverse image of each countably compact (compact) set is countably compact (compact) is said to be countably compact preserving__(comPact pres,erving[3 or co_rnpac in the terminoloqy of Liden [2] if image of each counLably compact(compact) set is countably compact(compact), is said to be perfect [2] if it is continuous,closed, and has compact fibers f-l(y), y E Y. X is said to be a Frgchet space if for each subset A of X0 x E; cl(A) implies there exists a sequence {Xn} in A converging to x X is said to be a k-space if O is open (equivalently:closed) in X whenever O K is open(closed) in K for every compact subset K of X Every Frchet space as well as every locally compact space is a k-space.
2. PROPOSITIONS.PROPOSITION 2.1.Let X--.Y be countably compact preserving(in particular, continuous), where X is cou.tablycompact and Y is a Frechet space Then is closed.PROOF.The proof is the same as that of Theoreln 16.19 of ThroI [22].PROPOSITION 2.2.Let f:X--,Y be closed with compact (countably compact) fibers,where X,Y are arbitrary.Then is compact (countably compact).PROOF.For the compact version, see Theorem 3 of Liden [2].His proof is valid for the non surjective case too.The proof of the other part is similar.
PROPOSITION 2.3.Let f:X-,Y be countably compact, where X is a Frchet space and Y is countably compact.Then is continuous.
PROOF.Let F be a closed subset of Y. Then F is countably compact.Since is countably compact,f "1 (F) is countably compact and so closed subset of X0 by Theorem 3.6 of Dugundji [21].Hence is continuous.
Combining propositions 2.2 and 2.3, we get the following: PROPOSITION 2.4.Let f:X-Y be closed with countably compact fibers, where X is a Fr$chet space and Y is countably compact.Then ts continuous.
COROLLARY 2.5.Let f:X--,, Y be a closed injection(bijection), where X is a Frchet space and Y is countably compact.
Then is an embedding(homeomorphism).
PROPOSITION 2.6.For any space X, let f:X--,Y be compact, where Y is a k-space.Then is closed.PROOF.For proof, see Theorem 2 of Liden [2].His proof is valid for the non surjective case too.PROPOSITION 2.7.For any space Y, let f:X-Y be compact preserving with closed fibers, where X is a k-space.Then is continuous.
PROOF.For proof, see Theorem 4 of Liden [2].His proof is valid for the non surjective case too.

THEOREMS.
The following theorems 3.1 and 3.3 give characterizations of perfect maps in compact and countably compact spaces, respectively.THEOREM 3,1.Let f:X Y be any map, where X,Y are compact spaces.Then the following are equivalent.(a) is perfect.
(h) is countably compact preserving NOTE/.Theorem 3.3 and corollary 3.4 donor hold even for compact spaces X and Y which are non-Frchet, as the following example shows.EXAMPLE 3.S.Let X Y [0, be the ordinal space, where is the first uncountable ordinal.Then the bijection

2 .
Let f:X---Y be any injection (bijection), where X,Y are compact spaces.Then the following are equivalent.(a) Is an embedding(homeomorphism).(b) is continuous.(c) is closed(or open).(d) has closed graph.(e) is compact.
(a) is perfect.(b) is continuous.(c) is closed.(d) has closed graph.(e) is compact.(f) is compact preserving.