space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→". See for example Fig. 4; there, the arrow from "real linear topological" to "real linear" is two-headed, since every real linear space admits some (at least one) topology compatible with its linear structure. Such topology is non-unique in general, but unique when the real linear space is finite-dimensional. For these spaces the transition is both injective and surjective, that is, bijective; see the arrow from "finite-dim real linear topological" to "finite-dim real linear" on Fig. 4. The inverse transition exists (and could be shown by a second, backward arrow). The two species of structures are thus equivalent. In practice, one makes no distinction between equivalent species of structures. Equivalent structures may be treated as a single structure, as shown by a large box on Fig. 4. The transitions denoted by the arrows obey isomorphisms. That is, two isomorphic lead to two isomorphic . The diagram on Fig. 4 is commutative. That is, all directed paths in the diagram with the same start and endpoints lead to the same result. Other diagrams below are also commutative, except for dashed arrows on Fig. 9. The arrow from "topological" to "measurable" is dashed for the reason explained there: "In order to turn a topological space into a measurable space one endows it with a σ-algebra. The σ-algebra of Borel sets is the most popular, but not the only choice." A solid arrow denotes a prevalent, so-called "canonical" transition that suggests itself naturally and is widely used, often implicitly, by default. For example, speaking about a continuous function on a Euclidean space, one need not specify its topology explicitly. In fact, alternative topologies exist and are used sometimes, for example, the fine topology; but these are always specified explicitly, since they are much less notable that the prevalent topology. A dashed arrow indicates that several transitions are in use and no one is quite prevalent. Types of spaces Linear and topological spaces Two basic spaces are linear spaces (also called vector spaces) and topological spaces. Linear spaces are of algebraic nature; there are real linear spaces (over the field of real numbers), complex linear spaces (over the field of complex numbers), and more generally, linear spaces over any field. Every complex linear space is also a