Structure of The Set of Liouville Numbers
For each positive integer n, set
.
The set of all Liouville numbers can thus be written as .
Each is an open set; as its closure contains all rationals (the {p/q}'s from each punctured interval), it is also a dense subset of real line. Since it is the intersection of countably many such open dense sets, is comeagre, that is to say, it is a dense Gδ set.
Along with the above remarks about measure, it shows that the set of Liouville numbers and its complement decompose the reals into two sets, one of which is meagre, and the other of Lebesgue measure zero.
Read more about this topic: Liouville Number
Famous quotes containing the words structure of, structure, set and/or numbers:
“Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.”
—Germaine Greer (b. 1939)
“Agnosticism is a perfectly respectable and tenable philosophical position; it is not dogmatic and makes no pronouncements about the ultimate truths of the universe. It remains open to evidence and persuasion; lacking faith, it nevertheless does not deride faith. Atheism, on the other hand, is as unyielding and dogmatic about religious belief as true believers are about heathens. It tries to use reason to demolish a structure that is not built upon reason.”
—Sydney J. Harris (19171986)
“The question is not ... if art is enough to fulfill my life, but if I am true to the path I have set for myself, if I am the best I can be in the things I do. Am I living up to the reasons I became a singer in the first place?”
—Kathleen Battle (b. 1948)
“Our religion vulgarly stands on numbers of believers. Whenever the appeal is madeno matter how indirectlyto numbers, proclamation is then and there made, that religion is not. He that finds God a sweet, enveloping presence, who shall dare to come in?”
—Ralph Waldo Emerson (18031882)