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MAPPINGS OF THE INFINITE-DIMENSIONAL SPACES
Ivan Lončar
Sažetak
Let f : X-^ Y be a mapping between normal spaces. Under which
conditions on f, X and Y does the A-weak (A-strong) infinite-dimensionality of Y y hold from A-weak (A-strong) infinite-dimensionalv of X.
In the present paper the partial answers for open and for closed
mappings are given.
In Section 1. we prove that, if f : X ^Y Is an open surjection
between separable metric spaces such that Frf—Hy) has an isolated
point (f~Hy) IS discrete), then Y is Aweakly infinite-dimensional if
(iff) X is A-veakly infinite-dimensional (1.4. Lemma and 1.13. Theorem). By virtue of this statements we prove that lim ?C is A-strongly infinite-dimensional if X is an inverse sequence of A-strongly
infinite-dimensional separable metric spaces such that 1 ^ | Frfnm
(xn) 1 ^ k (L8. Theorem). If Xis a well-ordered inverse system, then
one can assume that FrfTp (xa) is a discrete subspace of X3 (Theorem 1.10.). In Lemma 1.24. and 1.25. we prove that if f : X ^ Y is a local homeomorphism between hereditarily Lindelöf spaces, then a closed subset A of X is A-weakly infinite-dimensional iff f(A) is A-weakly infinite-dimensional. In theorems 1.28.—1.34. we consider the
inverse systems of A-weakly (A-strongly) infinite-dimensional spaces
and open bonding mappings with finite fibers. In this case the
limit of inverse system of infinite-dimensional Cantor-manifolds Is
an infinite-dimensional Cantor-manifold (Theorem 1. 35. and 1.36.).
Section 2. contains the results on infinite-dimensionality under
closed mappings with finite or contable fibers. Using Theorem 2.13.
and S-weak infinite-dimensionality of ßX (Lemma 2.28.) we prove
some theorems for a A-weak (A-strong) infinite^dlmensionality of a
limit of an inverse systems of countably compact spaces (2.22. Theorem) and for a a-directed inverse system of Lindelöf spaces (2.25. Theorem). Inverse systems of infinite-dimensional Cantor-manifolds are also considered (2.29. and 2.30. Theorem).
Ključne riječi
Hrčak ID:
133927
URI
Datum izdavanja:
5.12.1986.
Posjeta: 938 *