Advanced Calculus: Difference between revisions

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===Closed===
===Closed===
The folllowing definitions of Closed Sets are equivalent.
The folllowing definitions of Closed Sets are equivalent.
* (Order)
* (Order) A set <math>S</math> is open if for every number <math>x \in S</math>, there is an epsilon <math>\epsilon > 0</math> such that the epsilon ball is a subset of s: <math>\{ y | d(x, \epsilon) \} \subset S</math>. The compliment of an open set is closed.
* (Sequences) A set <math>S</math> is clsoed if it contains all its limit points. That is <math>\forall \{x_i\} \subseteq S</math>, <math>\{x_i\} \rightarrow x_0 \implies x_0 \in S</math>.
* (Sequences) A set <math>S</math> is closed if it contains all its limit points. That is <math>\forall \{x_i\} \subseteq S</math>, <math>\{x_i\} \rightarrow x_0 \implies x_0 \in S</math>.
* (Topology)


; Notes
; Notes