|
|
Thick sets in Banach spaces
Skříšovský, Emil ; Spurný, Jiří (advisor) ; Kalenda, Ondřej (referee)
This thesis studies thick sets in Banach spaces, which are defined similarly as the sets of the second Baire category. We show some basic properties of thick and thin sets and their characterizations - mainly the relation with the Uniform Boundedness Principle, the Banach-Steinhaus and Open mapping theorem and w∗ - integrability. Lastly, we give an example of a thick set, which is not of the second Baire category. 1
|
|
|
Dense sets in products of topological spaces
Bartoš, Adam ; Simon, Petr (advisor) ; Hušek, Miroslav (referee)
A subset of a product space is thin if every two its distinct points are distinct in at least two coordinates. A subset of a product space is very thin if every two its distinct points are distinct in all coordinates. The thesis sum- marizes the basic properties of thin-type dense sets in products of topological spaces. Sufficient and necessary conditions of their existence are given and several examples are shown. The main result of the thesis is a construction showing that under the continuum hypothesis, for every natural n ≥ 1, there exists a countable T3 dense-in-itself space X such that Xn contains an n-thin dense subset, but Xm , n < m < 2n, doesn't. Besides, Xm , n < m < ω, does not contain any (n + 1)-thin dense subset. A weaker form of the theorem is proven under Martin's axiom.
|
guest ::
