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A topology is a family ℐ of sets which satisfies the following conditions: (i) the empty set is a member of ℐ, (ii) the intersection of any two members of ℐ is a member of ℐ, (iii) the union of each subfamily of ℐ is a member of ℐ.

This paper deals with the generalization of the density points with respect to category. Topologies using this concept of density are introduced, and their properties are investigated.

Simple structure theorems or representation theorems make their appearance at a very elementary level in subjects such as algebra, but one infrequently encounters such theorems in elementary topology. In this note we offer one such theorem, for finite topological spaces, which involves only the most elementary concepts: discrete space, indiscrete space, product space and homeomorphism. Our theorem describes the structure of those finite spaces which are homogeneous.

Introduction to Topology By Prof. Bert Mendelson. Pp. ix + 217. (London and Glasgow: Blackie and Son, Ltd., 1963.) 35s. net.

The notion of a cellular complex which is well known in the topology is applied to describe the structure of images. It is shown that the topology of cellular complexes is the only possible topology of finite sets. Under this topology no contradictions or paradoxes arise when defining connected subsets and their boundaries. Ways of encoding images as cellular complexes are discussed. The process of image segmentation is considered as splitting (in the topological sense) a cellular complex into blocks of cells. The notion of a cell list is introduced as a precise and compact data structure for encoding segmented images. Some applications of this data structure to the image analysis are demonstrated.

For discrete groups we introduce a new class of sets, called weak uniqueness sets, which for abelian groups contains the class of sets of uniqueness. Considered is the problem of determining groups for which every finite set is a weak uniqueness set. Some examples are given.