139

normality is shown to be equivalent to the equaHty of A* n, B* and {A г\ В)*, for any two closed sets A, В oi the space, A* denoting the closure in the Öech-compactifieation. (Note tha-t jVicinity a* is used for ,entourage F» de diagonal' in the sense of A. Weil).

V . 8. Krishnan.

Isbell , J. R.: Homogeneous spaces. Duke math. J. 20, 321—329 (1953).

The relation between various types of homogeneity of topological spaces are first gated. A space is termed an s-spaee provided any two of its points have neighborhood bases which may be put into one-one correspondence so that corresponding terms are homeomorphic; the space is microhomogeneous if any two of its points x and у have homeomorphic neighborhoods under a map sending x toy. A space is almost homogeneous provided for any two points x and у there exists a homeomorphism of the space into itself sending x to y; the space is shrinkable about the point x provided there exists for each neighborhood Ü oi x a, homeomorphism of the space into Ü, leaving x fixed; it is shrinkable if it is shrinkable about all of its points. Inter^ting examples are given of: 1. a plane set which is an s-space, not almost homogeneous nor micro- homogeneous, locally coimected at and shrinkable about some but not all of its pointe; 2. a subset of the BHlbert cube which is shrinkable and almost homogeneous, but is not an «-space;

3 . a locally EucHdean space, made up of uncountably many planes, which is not involutory geneous. Further, it is shown that every microhomogeneous connected, hnearly ordered space is locally Birkhoff homogeneous; a linearly ordered space is Birkhoff homogeneous if it is order- isomorphic to аД its open intervals. —^ Eeviewer's remarks: a) The authors conjectiu-e tha| a shrinkable connected space is locally connected is false: let ^ be a homogeneous non locafly connected contuauum; according to the authors theorem 2.10, an iofioite power of Я is shrinkable, obviously connected but not locally connected, b) Exemple 3. above is intended to provide the negative answer to a problem of van Dantzig [Fundamenta Math. 15,102—125 (1930), Probl, 47] the answer is not very satisfactory since the authors locally EucHdean space has no countable base, whereas von Dantzig's manifolds are always assiuned simpUcial. T. Ganea,

Bing , E. H.: A connected countable Hausdorff space. Proc. Amer. math. Soc.

4 , 474 (1953).

Examples are constructed of connected countable Hausdorff spaces whose points are the rational points in a Euclidean half-plane, resp., in a quadrant of a JEuclidean three space, such that these are infinite-dimensional in the Lebesgue sense, but have dimension 1 and 2 respectively in the Menger-Ürysohn sense. The same method could be used to get similar spaces of any finite dimension in the Menger- Ürysohn sense. A correction that seems necessary in line 14 is to replace „ bourhoods" by „Open sets covering the space". F. S, Krishnan.

Yoneda , Nobuo; On the mappings of complexes. J, Fac. Sei., Univ. Tokyo, Sect. I 6, 393—419 (1953).

L'A . donne: 1° Une définition de l'isotopie combinatoire de deux sous- plexes d'un complexe au moyen de déformations locales dans l'étoile d'un simplexe ; il montre alors que deux applications simpliciales homotopes peuvent être déformées l'une dans l'autre par un produit de telles déformations. 2° Si f: К -> K' est une application simpliciale d'un complexe К de dimension m dans le complexe K' de dimension n, on appelle singularité de / sur un simplexe T^ de dimension q de K' l'excès sur q de la dimension du complexe image réciproque 1~^{Тч); de même, la „singularité" de / sur un sous-complexe A de K' est Sup des singularités de / sur les Simplexes de A. L'A. montre alors que si K' est un complexe qui a les propriétés d'homotopie locale d'une variété de dimension n, toute application f: К -^ K' peut être approchée à e près par une appUcation simpliciale /' pour laquelle la „singularité" est égale à Sup {m — %, 0). Application en est donnée à la théorie du degré, notamment à la théorie du degré tordu de l'apphcation canonique de ^ sur le plan projectif réel P^{R). B. Thym.

Samelson , Hans: A connection between the Whitehead and the Fontryagin product. Amer. J. Math. 75, 744—752 (1953).

Es sei X ein zusammenhängender, einfach-srasammenhängender Baum (л^(-ЗГ) = Щ.{Х) = 0; 1-zusammenhängend). Es sei Q{X) der Raum der g^chlc^senen Wege von X, die einen vor- g^ebenen Anfangs- und Endpunkt haben. Der Raum Q{X) besifczt eine natürliche kation: Wenn Wi,w^^Q(X), dann ist v\W2, der komponieaiie Weg (erat w^, dann w^). Diese Multiplikation CTzeugt eine Multiplikation für Homolc^elda^en a, b von ö (X) ; wenn а Ç Ж,(о(Х)) und Ъ ^Hg{Q (X)), dann ist а * ô С Я^« {Q{X)), (Pontrjaginsch^ Produkt). BekannÖich hat хааях