The central notion of continuity of functions is extended in this section to general topological spaces. The useful characterization of continuous functions in metric spaces as those functions where the inverse image of every open set is open is used as a definition in the general setting.
Because many properties of spaces are preserved by continuous functions, spaces related by a bijection (one-to-one and onto function) which is continuous in both directions will have many properties in common. These properties are identified as topological properties. Spaces so related are called homeomorphic.