Editing Accumulation point (section)

==Relation between accumulation point of a sequence and accumulation point of a set==

Every sequence <math>x_{\bull} = \left(x_n\right)_{n=1}^{\infty}</math> in <math>X</math> is by definition just a map <math>x_{\bull} : \N \to X</math> so that its [[Image (mathematics)|image]] <math>\operatorname{Im} x_{\bull} := \left\{ x_n : n \in \N \right\}</math> can be defined in the usual way. 

* If there exists an element <math>x \in X</math> that occurs infinitely many times in the sequence, <math>x</math> is an accumulation point of the sequence. But <math>x</math> need not be an accumulation point of the corresponding set <math>\operatorname{Im} x_{\bull}.</math> For example, if the sequence is the constant sequence with value <math>x,</math> we have <math>\operatorname{Im} x_{\bull} = \{ x \}</math> and <math>x</math> is an isolated point of <math>\operatorname{Im} x_{\bull}</math> and not an accumulation point of <math>\operatorname{Im} x_{\bull}.</math>

* If no element occurs infinitely many times in the sequence, for example if all the elements are distinct, any accumulation point of the sequence is an <math>\omega</math>-accumulation point of the associated set <math>\operatorname{Im} x_{\bull}.</math>

Conversely, given a countable infinite set <math>A \subseteq X</math> in <math>X,</math> we can enumerate all the elements of <math>A</math> in many ways, even with repeats, and thus associate with it many sequences <math>x_{\bull}</math> that will satisfy <math>A = \operatorname{Im} x_{\bull}.</math>

* Any <math>\omega</math>-accumulation point of <math>A</math> is an accumulation point of any of the corresponding sequences (because any neighborhood of the point will contain infinitely many elements of <math>A</math> and hence also infinitely many terms in any associated sequence).

* A point <math>x \in X</math> that is {{em|not}} an <math>\omega</math>-accumulation point of <math>A</math> cannot be an accumulation point of any of the associated sequences without infinite repeats (because <math>x</math> has a neighborhood that contains only finitely many (possibly even none) points of <math>A</math> and that neighborhood can only contain finitely many terms of such sequences).