Some results in general topology and applications to dynamical systems: Homogeneity and monotone semiflows

This thesis contains three chapters on unrelated topics from the fields of General Topology and Dynamical Systems.; It is proven that omega power of a first-countable zero-dimensional space is homogeneous. This answers a question of G. Gruenhage, posed as a New Classic Problem in Topology in 1990.; The Baire property is examined in the case of finite products of a space and its dense subspace.; The Theory of Monotone Semiflows is developed under weak order-preserving properties. This theory has already been developed by others under powerful montonicity conditions. Monotone Semiflows model an important class of continuous time dynamical systems. Set-theoretic examples are provided to show that these new order-preserving conditions are significantly weaker than the montonicity conditions previously studied.