We assume that two events may have a relation which I will call "compresence," which means, practically, that they overlap in space-time. Take, for instance, notes played by different instruments in orchestral music: if one is heard beginning before the other has ceased to be heard, the auditory percepts of the hearer have "compresence." If a group of events in one biography are all compresent with each other, there will be some place in space-time which is occupied by all of them. This place will be a "point" if there is no event outside the group which is compresent with all of them. We may therefore define a "point-instant," or simply a "point," in one biography, as a group of events having the following two properties: [Pg 295] (1) Any two members of the group are compresent; (2) No event outside the group is compresent with every member of the group. When we pass beyond one dimension, this method is no longer applicable. Take, for example, the three circles in the accompanying figure: each overlaps with the other two, but there is no region common to all three. If we try to remedy this (as I believe we can) by starting, in two dimensions, with a relation of three events, which is to hold when all three have a region in common, we are still met by difficulties. The three circles , , have a region in common, and the shaded area has a region in common with and , also with and , and also with and , yet , , and have no region in common. Therefore if events may have queer shapes such as , our new three-term relation will still not enable us to define a "point." [Pg 296] Since the problem with which we are concerned belongs to analysis situs, in which we are occupied only with such properties of figures as are unaffected by continuous deformation, we cannot simply declare in advance that no events are to have odd shapes. But before attempting to deal with this difficulty, it will be as well to consider certain points in analysis situs, which will show us what are the requisites of a solution of our problem. In analysis situs we start with two conceptions, that of a point, and that of "neighbourhoods of a given point"—the latter being collections of points. Certain definitions obtained in this way will be useful. The following definitions are due to Leopold Vietoris.[60] If is a set of points, a point is called a "Häufungspunkt" of if in every neighbourhood of there is a point other than . Two collections of points "touch" each other in a point if belongs to one collection and is a "Häufungspunkt" of the other. A set of points is "continuous from to " if it contains and , and any two parts of it whose sum is , of which one contains and the other , touch each other (in at least one point). A set of points is a "Linienstück" from to if it, but none of its proper parts, is continuous from to . Hausdorff[61] has defined a "metrical" space and a "topological" space in the following terms.