Some properties of four topologies in real linear spaces and three classes related to them

In the paper there are invesigated the core topology, ${\tau }_1$, the directional topology, ${\tau }_2$, the Klee topology, ${\tau }_3$, and the finite topology, ${\tau }_0$, as well as the generalizations ${\tau }_1(n)$, ${\tau }_2(n)$ and ${\tau }_3(n)$ of ${\tau }_1),\text{(}{\tau }_2$ and ${\tau }_3$, respectively. These generalizations are obtained when in the definition of a given topology the condition concerning straight lines is replaced by the analogous condition concerning linear varieties of dimension $n$, where $n\in \text{mathbbP}N$. There are stated the inclusions between these topologies, the characterization with the respect to separation axioms. There are answered the questions: when considered topological spaces are Baire, sequential and Fréchet? There are formulated criteria for the compactness and sequentially compactness of sets, for the convergence of sequences. There is stated the characterization of curves. Till now these problems were undertaken only in particular cases and for some topologies ${\tau }_1$, ${\tau }_2$, ${\tau }_3$ and ${\tau }_0$. For all considered topologies as well as for a certain class (including linear topologies) there are characterized the components of open sets; it is shown that every such component is the arcwise connected component and the quasi-component. In the paper there is also discussed the problem: what is it obtained when in the definition of the topology ${\tau }_2(n)$ instead of $\mathbb{R}$ there is an other topological space?