| Popis: |
We connect Whitney levels and continuous functions between continua to obtain the new notion of Whitney preserving maps. We introduce the basic properties of Whitney preserving maps. We give conditions on a continuum X in order that a Whitney preserving map f from X to the unit interval is a homeomorphism, and we give examples to show the conditions are necessary and that the result is false when the range of the map is the unit circle. Concerning the structure of any continuum X, we show that if A is a continuous decomposition of X into nondegenerate terminal continua, then there is a Whitney preserving map from X to the quotient space X/A. We also show that if f : X ? Y is a Whitney preserving map, then the highest Whitney level of C(X) mapping to the zero Whitney level of C(Y) is a continuous decomposition of X into terminal continua. We introduce the notion of a strictly Whitney preserving map; we show that being strictly Whitney preserving is equivalent to being hereditarily irreducible when the map is weakly confluent. |
