Robert Cauty

Open problems in infinite-dimensional topology

Publisher Summary This chapter discusses open problems in infinite-dimensional topology. The development of infinite-dimensional topology was greatly stimulated by three famous open problem lists—that of Geoghegan, West and Dobrowolski, Mogilski. It is now expected that the future progress will happen on the intersection of infinite-dimensional topology with neighbor areas of mathematics—Dimension Theory, Descriptive Set Theory, Analysis, and Theory of Retracts. This chapter attempts to select the problems whose solution would require creating new methods. It defines a pair as a pair (X, Y) consisting of a space X and a subspace Y ⊂ X. The ω denotes the set of non-negative integers. Many results and objects of infinite-dimensional topology have zero-dimensional counterparts usually considered in Descriptive Set Theory. As a rule, “zero dimensional” results have simpler proofs compared to their higher dimensional counterparts. Some zero-dimensional results are proved by essentially zero-dimensional methods (like those of infinite game theory), and it is an open question to which extent their higher-dimensional analogues are true. The chapter elaborates about the higher dimensional descriptive set theory, concepts of Zn-sets, along with presenting the related questions..

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2X is homeomorphic to Π2 ifX is a Baire space, and toQ∖Π1 ifX contains a dense absoluteGδ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π1, Π1⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes.

Un théorème de point fixe pour les fonctions multivoques acycliques

Abstract We prove that equiconnected spaces have the fixed point property for multivalued u.s.c. acyclic compact mappings.

L'espace des plongements d'un arc dans une surface

We prove that the space of embeddings of an arc into a surface without boundary M is homeomorphic to the product U(M) × l 2 , where U(M) is the unit tangent bundle of M