## Conditions under which disks are $P$-liftable

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- by Edythe P. Woodruff PDF
- Trans. Amer. Math. Soc.
**186**(1973), 403-418 Request permission

## Abstract:

A generalization of the concept of lifting of an*n*-cell is studied. In the usual upper semicontinuous decomposition terminology, let

*S*be a space, $S/G$ be the decomposition space, and the projection mapping be $P:S \to S/G$ . A set $Xβ \subset S$ is said to be a

*P*-lift of a set $X \subset S/G$ if $Xβ$ is homeomorphic to

*X*and $P(Xβ)$ is

*X*. Examples are given in which the union of two

*P*-liftable sets does not

*P*-lift. We prove that if compact 2-manifolds

*A*and

*B*each

*P*-lift, their union is a disk in ${E^3}/G$, their intersection misses the singular points of the projection mapping, and the intersection of the singular points with the union of

*A*and

*B*is a 0-dimensional set, then the union of

*A*and

*B*does

*P*-lift. Even if a disk

*D*does not

*P*-lift, it is proven that for $\epsilon > 0$ there is a

*P*-liftable disk $\epsilon$-homeomorphic to

*D*, provided that the given decomposition of ${E^3}$ is either definable by 3-cells, or the set of nondegenerate elements is countable and ${E^3}/G$ is homeomorphic to ${E^3}$. With further restrictions on the decomposition, this

*P*-liftable disk can be chosen in such a manner that it agrees with

*D*on the singular points of

*D*.

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*Concerning the condition that a disk in*${E^3}/G$

*be the image of a disk in*${E^3}$, Doctoral Dissertation, SUNY/Binghamton, 1971. β,

*Examples of disks in*${E^3}/G$

*which can not be approximated by P-liftable disks*(to appear).

## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**186**(1973), 403-418 - MSC: Primary 57A10; Secondary 54B15
- DOI: https://doi.org/10.1090/S0002-9947-1973-0328943-6
- MathSciNet review: 0328943