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Derived set (mathematics)
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==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 <math>X</math> can be equipped with an operator <math>S \mapsto S^*</math> mapping subsets of <math>X</math> to subsets of <math>X,</math> such that for any set <math>S</math> and any point <math>a</math>: # <math>\varnothing^* = \varnothing</math> # <math>S^{**} \subseteq S^*\cup S</math> # <math>a \in S^*</math> implies <math>a \in (S \setminus \{a\})^*</math> # <math>(S \cup T)^* \subseteq S^* \cup T^*</math> # <math>S \subseteq T</math> implies <math>S^* \subseteq T^*.</math> <!-- The following is wrong, see discussion page, section "S** subset S*" 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: # <math>\varnothing^* = \varnothing</math> # <math>S^{**} \subseteq S^*</math> *3'. <math>S^* = (S \setminus \{a\})^*</math> *4'. <math> \, (S \cup T)^* = S^* \cup T^*</math> --> Calling a set <math>S</math> {{em|closed}} if <math>S^* \subseteq S</math> will define a topology on the space in which <math>S \mapsto S^*</math> is the derived set operator, that is, <math>S^* = S'.</math>
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