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)
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