D-21 - Connectedness

Abstract

The geometrical figure was always treated by the Greeks as a continuum, although for physical substances the atomistic ideas prevailed. The final conclusion of Aristotle's reasoning was the claim that continuum cannot be composed of points. According to this conviction, Euclid's Elements were written. For Euclid, the points were nothing more than signs of places on the figure. The scholastic philosophers confirmed Aristotle's conviction, which was accepted by mathematicians till the nineteenth century. Only Dedekind definitively rejected Aristotle's objections concerning the point structure of continuum by constructing his arithmetical continuum composed of numbers that are now called real. Continuous images of connected spaces are connected. The union of two connected sets is connected if the sets have points in common. Maximal connected subsets are called components of the space. They are closed and pair wise disjoint. Hereditarily disconnected spaces contain, by definition, no non-trivial connected subsets. The class of spaces called punctiform is much wider and contains no non-trivial continua. A space is said to be extremally disconnected if the closures of open subsets are open.