A topological space $F$ consisting of two points. If both one-point subsets in $F$ are open (both are then closed), $F$ is said to be a simple colon. If only one one-point subset in $F$ is open, $F$ is said to be a connected colon. Finally, if only the empty subset and all of $F$ in $F$ are open, $F$ is called an identified colon; this space — unlike the first two, which are very important though simple — has found no applications.
|||P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)|
The term two-point discrete space is often applied to the simple colon. Its topological powers are called Cantor cubes. These spaces are universal in two ways: Every zero-dimensional compactum can be imbedded into a Cantor cube of the same weight, and every compactum can be obtained as the continuous image of a closed set of a Cantor cube of the same weight. These facts generalize well-known results on the Cantor cube of countable weight, the Cantor set. The connected colon is also known as Sierpiński space. Its topological powers are called Alexandrov cubes; they are universal in that they contain all $T_0$-spaces topologically.
|[a1]||R. Engelking, "General topology" , PWN (1977) (Translated from Polish)|
Colon. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Colon&oldid=37285