Finite sets are closed iff the first separation axiom takes place. Third and fourth separation axioms, regular and normal spaces. Metric spaces satisfy all the separation axioms. Second countability axiom. Separability and its relation to the second countability.
Euclidean spaces are separable.
First countability axiom. Sequential definitions of topology and continuous map. Inheritness of countability axioms. Compactness of a closed subset of a compact space. Compactness of a closed interval.