* Lebesgue's Fundamental Lemma (Pflastersatz) may be considered as a result of this kind, since it gives a local property based on assumptions about the dimension of a set in the large.

† This has led also to numerous attempts to approach the general set (or space) by means of local definitions of a restrictive character. The disadvantage of such an approach—unless it serves some practical purpose—is obvious: the object of our studies can easily become dependent on these definitions and the conditions expressed in them, which are mostly meaningless for general cases.

* See also my note in Comptes Rendus of Aug. 12, 1935 which appeared after the completion of this paper in June 1935.

† Definitions of infinite cycles (Fundamentalfolgen, Vollzyklen) are given in Vietoris', Lefschetz' and Alexandroff's papers, in which the main invariants (connectivity numbers, etc.) of a set are introduced.

* Alexandroff, P., “Untersuchungen über Gestalt und Lage abgeschlossener Mengen beliebiger Dimension”, Annals of Mathematics, 30 (1928), 101–187CrossRefGoogle Scholar; “Zum allgemeinen Dimensionsproblem”, Nachr. Ges. Wiss. Göttingen (1928), 25–44Google Scholar; “Dimensionstheorie” Math. Annalen 106 (1932), 161–238.Google Scholar

† This definition of a nucleus of a set differs formally from the original definition by Alexandroff, but follows directly from his results.

‡ A Cantor manifold in the case of the geometrical theory mod 2 is defined in the same way as in the case of Brouwer dimensions.

* Decreasing nuclei are defined of course rel. some decreasing spherical neighbourhoods in R ⁿ.

† In some recent papers (see Proc. Camb. Phil. Soc. 30 (1934), 428–452) Idefined manifold-points and Risspunkte as limitpoints of sequences of decreasing Cantor manifolds. The present definitions are not only more adequate for our purpose but lead to an obvious deepening of the results. The definitions given above can be directly generalized for the case of Brouwer dimensions.

‡ These results can be generalized for the case of sets of arbitrary dimension in Brouwer's sense. From the combinatory point of view the proofs involve an approximation of the sets by sequences of cycles with variable moduli, as shown by Alexandroff. In this case theorem 3 is valid also for Risspunkte. I shall deal with this case in another paper.

§ In the special case r = n − 1, namely when F is an (n − 1)-dimensional surface linked with a O-cycle and cutting the space, this theorem was proved in a joint paper by Ursell, H. D. and myself (Proc. Nat. Ac. Sc. 20 (1934), 662–666)Google Scholar. In this case a methodically quite different proof based on the Brouwer order of continuous representation of specially (“harmonically”) subdivided complexes was possible. But this method fails entirely in the general case of a set of arbitrary dimension.

* To construct such a division of z′_k we form a complex of z′_k with at least one 0-simplex on F′ and denote by z ^r−1 the common boundary of and By an infinitesimal shifting of all vertices of z ^r−1 we can arrange that this cycle lies entirely in B. In this way will be obtained in the required form.

* We can obtain K ^{n−r + 1} by joining up a point inside U in a general position with respect to Λ^n−r to all points of Λ^n−r by straight segments. Then all i-simplices of Λ^n−r will be in (1, 1) correspondence with (i + 1)-simplices of K ^n−r+1.

* Similar (n − r)-dimensional complexes Q ^n−r with respect to F were constructed by Alexandroff, who calls such a construction an “ε-modification” of a complex. In our case we have to construct an (n − r + 1)-complex with the described properties with respect to F and B simultaneously. For n − r + 1 = 2 similar complexes were constructed (under the name of “dual membranes”) quite independently in papers by H. D. Ursell and by myself.

† This complex can be assumed to lie inside any n-sphere containing the cycle

‡ "ε-shifting” (ε-Verschiebung) of a given complex into a complex K′ is a transformation under which all vertices of K and K′ are in (1, 1) correspondence and two corresponding vertices lie within an (arbitrarily small) distance ε.

§ The proof of this homology is very easy, see Alexandroff, “Dimensionstheorie”, § 4, where a short proof is given, based on a method due to J. W. Alexander.

∥ In fact we could carry out all the constructions without affecting z^n-r-1 and make *z^n-r-1=z^n-r-1

† The complexes ξⁱ of different dimension can be constructed by induction in the manner described in § II.

‡ Since the distance ρ of the sets and is greater than O' we can construct so that each corresponding pair ζ^n-r and ζ^n-r will lie within a distance less than ⅓ρ.