We study the space of complete Riemannian metrics of nonnegative curvature on the sphere equipped with C^{k+\alpha} topology. We show the space is homogenous for k>=2. If k is infinite, we show that the space is homeomorphic to the separable Hilbert space. We also prove for finite k, the space minius any compact subset is weakly contractible.

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We study the complexity of the homeomorphism relation on the classes of metrizable compacta and Peano continua using the notion of Borel reducibil- ity. For each of these two classes we consider two different codings. Metrizable compacta can be naturally coded by the space of compact subsets of the Hilbert cube with the Vietoris topology. Alternatively, we can use the space of continuous functions from the Cantor space to the Hilbert cube with the topology of uniform convergence, where two functions are considered as equivalent iff their images are homeomorphic. Similarly, Peano continua can be coded either by the space of Peano subcontinua of the Hilbert cube, or (due to the Hahn-Mazurkiewicz theo- rem) by the space of continuous functions from r0, 1s to the Hilbert cube. We show that for both classes the two codings have the same complexity (the complexity of the universal orbit equivalence relation). Among other results, we also prove that the homeomorphism relation on the space of nonempty compact subsets of any given Polish space is Borel bireducible with the above mentioned equivalence relation on the space of continuous functions from the Cantor space to the Polish space.

This thesis deals with study of mappings preserving Borel classes or absolute Borel classes. We prove a theorem which shows that under some assumptions there exists a (selection) function with certain properties. Using this theorem we obtain several results on preservation of Borel classes. Moreover, thanks to that theorem we prove a theorem on preservation of absolute Borel classes under a perfect mapping. Next, we show an assertion which implies that a piecewise closed mapping has a restriction that is "piecewise perfect" and its image is equal to the image of the original mapping. Under certain additional assumptions we prove a similar assertion for an Fσ-mapping instead of a piecewise closed mapping. Using these assertions and the theorem on preservation of absolute Borel classes under a perfect mapping we obtain further results on preservation of absolute Borel classes, in particular, for piecewise closed mappings and Fσ- -mappings. In the last chapter we study mappings such that the inverse image of an open set under these mappings is of a particular additive class. 1