ANALYSIS SITUS (POSITION ANALYSIS). In this branch of mathematics shape and size are unimportant. The im portant property may be said, roughly, to be proximity; to state it in precise terms, we must introduce the idea of limit point. Let S denote the collection of all points in the Euclidean plane (i.e., the plane studied in Euclidean plane geometry). Then a point, P, is called a limit point of a collection of points M if every circle whose centre is at P encloses at least one point of M which is distinct from P. It is easily seen from this definition that every point of S is a limit point of S. Also every point of a line is a limit point of the collection of all points on that line. Similar statements may be made for all the ordinary figures of plane geometry. In Analysis situs we are interested only in those prop erties of figures which can be expressed in terms of collections, or sets, of points and their limit points—such properties are called the topological properties of the figures. A geometrical figure is regarded more in the sense of being a point set than as being a "figure." (See POINT SETS.) From another point of view it may be said that Analysis situs is interested only in those properties of figures or point sets which are not changed when the figures are subjected to continuous motion in a fluid medium. Thus, it is easy to conceive of a fluid motion carrying a circle into an ellipse and thence into a square, even though the areas enclosed by these figures differ greatly. From the standpoint of Analysis situs, then, there is no essential difference between these figures—the differences in their sizes and shapes are unimportant. However, the idea of continuous fluid motion is not precise enough for an exact characterization of Analysis situs. In order to make the idea implied by the illustra tion more exact, we introduce the idea of homeomorphism, and here we are led again to the idea of limit point.
Homeomorphism.—Twopoint sets, A and B, are said to be homoeomorplzic if their points can be made to correspond in a one to-one manner, those of A to those of B, so that if A, is any set of points in A having a point in A as a limit point, then the set, of corresponding points in B has the point P, of B which corre sponds to as a limit point, and conversely. In figure I consider the sets A, B and C. In A the set of points consisting of a,, ..., that is, all a's having odd subscripts, has a as a limit point (we suppose that there are infinitely many points in A, the sub scripts of the a's running through the entire set of positive in tegers) ; similarly the set of a's with even subscripts has a as a limit point. It may be said that these two sets converge to a from two opposite directions. In B there are three sets of points, con sisting of (I) ... (2) b,, b,, ..., and (3) b,, ..., converging to b from three different directions. The set C contains a set of points consisting of ..., which has two limit points, di and d,. Now A and B are homeomorphic, since by simply mak ing every correspond to the of the same subscript, and a to b, the definition is satisfied. But neither A nor B is homeo morphic with C, for no correspondence can be set up between A and C, say, which will satisfy the definition. If both d, and d, were made to correspond to a, then the latter part of the definition would be satisfied, but then the correspondence would not be one-to-one.
It can be seen that squares, circles, ellipses, triangles and poly gons are all equivalent in the sense that they are homeomorphic. And if we substitute the precise notion of homeomorphism for that of fluid motion, we can say that Analysis situs is interested only in those properties which are common to all figures that are homeomorphic. It might be asked, after observing that the geo metric figures just mentioned do not have a common shape or a common size, and hence seem to have very little in common, just what properties are left to be studied? It may seem that there is very little left to be studied. To show that this is by no means the case, will be the purpose of the remainder of this article. We shall not attempt to cover all of the various aspects of Analysis situs, but take up only sufficient of the elementary notions to accomplish our purpose.
Connectedness. A glance at the various familiar figures of plane geometry, such as the line, the triangle, the circle and the polygon, impresses one with a general property of such collections of points, namely their connectedness. Thus, a straight line, al though it is made up of points, is somehow to be thought of as a connected thing, in the sense that if it were possible to pick up the line with one's hands it would not fall apart, but would hang together like a piece of twine. It was an early problem of Analysis situs to express in some precise way just what this connectedness of sets of points is. The most successful attempt at this is em bodied in the so-called Lennes-Hausdorff definition (I) (2) . A point set M is connected if it is impossible to separate it into two sets A and B such that A and B have no common points and such that neither A nor B contains a limit point of the other. It can be shown that all of the geometrical figures referred to above are connected according to this definition. Thus, no matter how we divide the set of all points on a line into two groups, at least one of these will contain at least one limit point of the other. It is easy to prove that if two point sets are homeomorphic and one of them is connected, then the other is connected ; i.e., connectedness is a common property of homeomorphic point sets.
Recent researches have demonstrated, however, that the con nectedness implied by this definition is not as strong as the "con nectedness" one has in mind when thinking of a line as the geometric analogue of a piece of twine. If a point is removed from a line, the remainder is separated into two portions each of which is itself connected. If we remove a point from a circle what is left is still connected; i.e., no point of a circle disconnects the circle. But the existence has been established (3) (4) of certain sets of points in S, which, though they are connected according to the above definition, can be totally disconnected by the omission of single points. That is, there exists, in S, a connected point set M containing a certain point P which, if omitted from M, leaves no connected portion in M whatsoever.
It is natural for the mathema tician, having generalized from the figures of geometry to the continuum, to investigate the nature and possible structure of continua from a topological standpoint. Perhaps the simplest continuum is the arc, which is a set of points that is homeomorphic with a straight line interval. Another simple continuum is the simple closed curve which is a set of points that is homeomorphic with a circle. Both arcs and simple closed curves are special cases of continuous curves, which form a much more general class of continua. In order to characterize those continua which are continuous curves we introduce the idea of local connectedness.
A continuous curve is a con tinuum which is locally connected at every one of its points. All of the figures studied in geom etry are continuous curves.
However, not all continua are continuous curves. In figure 3, let AA, be a straight line interval, whose length may be called I unit. Let A., be z unit from A on AA„ be -4 unit from A on and, in general, let A,,, where n is any positive integer, be at a distance I/„ unit from A and At every such point, erect a perpendicular, A„B,, to AA,, say i unit in length, and at A erect a perpendicular AB that is also i unit in length. Consider the set of points, M, consisting of all points on the straight line intervals AA,, AB, and B„ (for every n). The set M is a continuum. It is not a continuous curve, however, because it is not locally connected at all of its points. For let P be a point of AB not coincident with A. Let C be a circle of radius r less than the distance from A to P. Then, no matter what circle K is selected with centre P, that circle, K, will enclose a point, Q, of some A„B,, say and there will be no connected portion of M containing both P and Q and lying wholly within C.
For references concerning continuous curves, as well as for a description of some of their properties, the reader is directed to reference (5) of the bibliography at the end of this article. Here we shall close our discussion of those properties of point sets that are studied in Analysis situs, having selected only a few of the fundamental ones by way of illustration.
In a recent work (8) of a philosophical nature—a work which seems to contain much of value not only for the mathematician and physicist but for the scientist in general, and which discusses the relation between the world as it actually exists with the world as conceived by the physicist—it is interesting to note that the author has made use of quite recent results from the field of Analysis situs.