On the Topology of Grid Continua
نویسندگان
چکیده
One-dimensional and two-dimensional continua belong to the basic notions of settheoretical topology and represent a subfield of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic definitions and properties of grid continua in R2 and R3 are summarised. Particularly, simple one-dimensional grid continua in R2 and in R3, and simple closed two-dimensional grid continua in R3 are emphasised. Concepts for measuring the length of one-dimensional grid continua, or the surface area of two-dimensional grid continua are introduced and discussed. 1 Institute of Control Theory and Robotics, Slovak Academy of Sciences, Bratislava, Slovak Republic 2 CITR, Tamaki Campus, University Of Auckland, Auckland, New Zealand On the topology of grid continua Fridrich Sloboda , Bedrich Za tko , and Reinhard Klette 2 1 Institute of Control Theory and Robotics, Slovak Academy of Sciences D ubravsk a cesta 9, 842 37 Bratislava, Slovak Republic 2 CITR, University of Auckland Tamaki Campus, Building 731, Auckland, New Zealand Abstract: One-dimensional and two-dimensional continua belong to the basic notions of set-theoretical topology and represent a sub eld of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic de nitions and properties of grid continua in R and R are summarised. Particularly, simple one-dimensional grid continua in R and in R, and simple closed two-dimensional grid continua in R are emphasised. Concepts for measuring the length of one-dimensional grid continua and the surface area of a two-dimensional grid continuum are introduced and discussed. One-dimensional and two-dimensional continua belong to the basic notions of set-theoretical topology and represent a sub eld of the theory of dimensions developed by P. Urysohn and K. Menger. In this paper basic de nitions and properties of grid continua in R and R are summarised. Particularly, simple one-dimensional grid continua in R and in R, and simple closed two-dimensional grid continua in R are emphasised. Concepts for measuring the length of one-dimensional grid continua and the surface area of a two-dimensional grid continuum are introduced and discussed.
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