|
|
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 ::
