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- W2153228275 abstract "In this paper we define and discuss some of the properties of certain subsets of a continuous curve,t which we call the arc-curves. Probably the most interesting property of these subsets is that whenever the arc-curve is closed, it is itself a continuous curve. The arc-curves are intimately connected with the theory of separation of points in continuous curves, since if a closed set separates a pair of points in the arc-curve of the pair of points with respect to a given continuous curve, the closed set separates the points in the given continuous curve. We define a basic set of a continuous curve as any set whose arc-curve is the continuous curve itself. Necessary and sufficient conditions are given in order that a subset of a continuous curve be a basic set and in order that a basic set be irreducible. It is assumed that all point sets of the paper lie in a two-dimensional euclidean space.t Notation. Wherever a symbol H is used to denote a point set, the symbol H7 denotes the point set consisting of the points of H plus all limit points of H. If u and v are points of a continuous ae, the symbol uv denotes the subarc of ae with u and v as end points if ui v, and if u = v, the symbol uv denotes the single point u(=v). Hereafter we shall use the term arc to mean simple continuous arc. If uv is an with end points u and v, the symbols , denote uv-u, uv-v, uv-u-v respectively. If A and B are sets of points, the symbol A c B means A is a subset of B, but it is not implied that A is a proper subset of B." @default.
- W2153228275 created "2016-06-24" @default.
- W2153228275 creator A5047614401 @default.
- W2153228275 date "1928-03-01" @default.
- W2153228275 modified "2023-09-26" @default.
- W2153228275 title "Concerning the arc-curves and basic sets of a continuous curve" @default.
- W2153228275 doi "https://doi.org/10.1090/s0002-9947-1928-1501446-1" @default.
- W2153228275 hasPublicationYear "1928" @default.
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