term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré. His first article on this topic appeared in 1894. In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that a surface is a topological space that is locally like a Euclidean plane. Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory". Metric spaces had been defined earlier in 1906 by Maurice Fréchet, though it was Hausdorff who popularised the term "metric space" (). Definitions The utility of the concept of a topology is shown by the fact that there are several equivalent definitions of this mathematical structure. Thus one chooses the axiomatization suited for the application. The most commonly used is that in terms of , but perhaps more intuitive is that in terms of and so this is given first. Definition via neighbourhoods This axiomatization is due to Felix Hausdorff. Let be a set; the elements of are usually called , though they can be any mathematical object. We allow to be empty. Let be a function assigning to each (point) in a non-empty collection of subsets of The elements of will be called of with respect to (or, simply, ). The function is called a neighbourhood topology if the axioms below are satisfied; and then with is called a topological space. If is a neighbourhood of (i.e., ), then In other words, each point belongs to every one of its neighbourhoods. If is a subset of and includes a neighbourhood of then is a neighbourhood of I.e., every superset of a neighbourhood of a point is again a neighbourhood of The intersection of two neighbourhoods of is a neighbourhood of Any neighbourhood of includes a neighbourhood of such that is a neighbourhood of each point of The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together the neighbourhoods of different points of A standard example of such a system of neighbourhoods is for the real line where a subset of is defined to be a of a real number if it includes an open interval containing Given such a structure, a subset of is defined to be open if