selected mathematical objects (for instance, functions on another space, or subspaces of another space, or just elements of a set) treated as points, and selected relationships between these points. Therefore, spaces are just mathematical structures of convenience. One may expect that the structures called "spaces" are perceived more geometrically than other mathematical objects, but this is not always true. According to the famous inaugural lecture given by Bernhard Riemann in 1854, every mathematical object parametrized by n real numbers may be treated as a point of the n-dimensional space of all such objects. Contemporary mathematicians follow this idea routinely and find it extremely suggestive to use the terminology of classical geometry nearly everywhere. Functions are important mathematical objects. Usually they form infinite-dimensional function spaces, as noted already by Riemann and elaborated in the 20th century by functional analysis. Taxonomy of spaces Three taxonomic ranks While each type of space has its own definition, the general idea of "space" evades formalization. Some structures are called spaces, other are not, without a formal criterion. Moreover, there is no consensus on the general idea of "structure". According to Pudlák, "Mathematics [...] cannot be explained completely by a single concept such as the mathematical structure. Nevertheless, Bourbaki's structuralist approach is the best that we have." We will return to Bourbaki's structuralist approach in the last section "Spaces and structures", while we now outline a possible classification of spaces (and structures) in the spirit of Bourbaki. We classify spaces on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties? This leads to the first (upper) classification level. On the second level, one takes into account answers to especially important questions (among the questions that make sense according to the first level). On the third level of classification, one takes into account answers to all possible questions. For example, the upper-level classification distinguishes between Euclidean and projective spaces, since the distance between two points is defined in Euclidean spaces but undefined in projective spaces. Another example. The question "what is the sum of the three angles of a triangle" makes sense in a Euclidean space but not in a projective space. In a non-Euclidean space the question makes sense but is answered differently, which is not an upper-level distinction. Also, the distinction between a Euclidean plane and a Euclidean 3-dimensional space is