Derived Set (mathematics) - Topology in Terms of Derived Sets

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.