On convergent sequences and properties of subspaces

We consider problems connected with the notion of convergent sequences in $T_1$-spaces. The properties of $T_1$-spaces and convergent sequences in these spaces considerably differ from the same properties of Hausdorff spaces. We consider problems connected with the properties of the minimal $T_1$-space. We consider properties of spaces where every sequence is a convergent sequence (Theorems 1 and 2 and Example 1). One of the main problems is the connection between convergent sequences and the properties of subspaces of the space. It is well known that the compactness, countable compactness and sequential compactness are not equivalent in general. We prove (Theorem 7) that hereditary sequential compactness, compactness and countable compactness are equivalent.