Topology in Terms of Derived Sets
Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. A set of points X can be equipped with an operator * mapping subsets of X to subsets of X, such that for any set S and any point a:
Note that given 5, 3 is equivalent to 3' below, and that 4 and 5 together are equivalent to 4' below, so we have the following equivalent axioms:
- 3'.
- 4'.
Calling a set S closed if will define a topology on the space in which * is the derived set operator, that is, . If we also require that the derived set of a set consisting of a single element be empty, the resulting space will be a T1 space. In fact, 2 and 3' can fail in a space that is not T1.
Read more about this topic: Derived Set (mathematics)
Famous quotes containing the words terms, derived and/or sets:
“A radical is one of whom people say “He goes too far.” A conservative, on the other hand, is one who “doesn’t go far enough.” Then there is the reactionary, “one who doesn’t go at all.” All these terms are more or less objectionable, wherefore we have coined the term “progressive.” I should say that a progressive is one who insists upon recognizing new facts as they present themselves—one who adjusts legislation to these new facts.”
—Woodrow Wilson (1856–1924)
“The sceptics assert, though absurdly, that the origin of all religious worship was derived from the utility of inanimate objects, as the sun and moon, to the support and well-being of mankind.”
—David Hume (1711–1776)
“A continual feast of commendation is only to be obtained by merit or by wealth: many are therefore obliged to content themselves with single morsels, and recompense the infrequency of their enjoyment by excess and riot, whenever fortune sets the banquet before them.”
—Samuel Johnson (1709–1784)