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National Repository of Grey Literature 2 records found  Search took 0.01 seconds. 
Borel sets in topological spaces
 Vondrouš, David ; Spurný, Jiří (advisor) ; Holický, Petr (referee)
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
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Continuous mappings and fixed-point theorems
 Vondrouš, David ; Holický, Petr (advisor) ; Zelený, Miroslav (referee)
This thesis deals with images of compact convex sets under a continuous mapping. We will show a combinatorial proof of famous Brouwer's fixed-point theorem based on Sperner's lemma. Later, this theorem will be applied for proving Brouwer's invariance of domain theorem, which asserts that image of an open subset of an euclidean space under a continuous mapping is open too. Then we will compare this proof with another proof using Borsuk's theorems. Their proof is more complicated, nevertheless it turns out that Borsuk's theorems give stronger results. One of them is, for instance, an analogy of the Darboux property for continuous mappings in an multidimensional space. 1
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