User:IssaRice/Adherent point and limit point: Difference between revisions

Revision as of 04:27, 6 July 2019

Let $(X, d)$ be a metric space, let $E$ be a subset of $X$ , and let $x_{0} \in X$ be a point.

there exists a sequence $(x_{n})_{n = 1}^{\infty}$ of points in $E$ which converges to $x_{0}$
for every radius $r > 0$ the ball $B (x_{0}, r)$ has nonempty intersection with $E$
$x_{0}$ is an interior point of $E$ or is a boundary point of $E$
for every open set $U$ such that $x \in U$ one has $U \cap E \neq \emptyset$
for every open set $U$ such that $x \in U$ there is some $y \in U \cap E$ such that $y \neq x$
for every open set $U$ such that $x \in U$ , the set $U \cap E$ has infinitely many points
there exists a sequence $(x_{n})_{n = 1}^{\infty}$ of distinct points in $E$ (i.e. $x_{n} \in E$ for all $n \geq 1$ and $x_{n} \neq x_{m}$ for all $n \neq m$ ) which converges to $x_{0}$
there exists a sequence $(x_{n})_{n = 1}^{\infty}$ of points in $E$ , none of which are equal to $x_{0}$ , which converges to $x_{0}$

@@ Line 4: / Line 4: @@
 * for every radius <math>r > 0</math> the ball <math>B(x_0,r)</math> has nonempty intersection with <math>E</math>
 * <math>x_0</math> is an interior point of <math>E</math> or is a boundary point of <math>E</math>
-* for every open set <math>U</math> such that <math>x \in U</math> one has <math>U\cap A \ne \emptyset</math>
+* for every open set <math>U</math> such that <math>x \in U</math> one has <math>U\cap E \ne \emptyset</math>
-* for every open set <math>U</math> such that <math>x \in U</math> there is some <math>y \in U\cap A</math> such that <math>y \ne x</math>
+* for every open set <math>U</math> such that <math>x \in U</math> there is some <math>y \in U\cap E</math> such that <math>y \ne x</math>
-* for every open set <math>U</math> such that <math>x \in U</math>, the set <math>U \cap A</math> has infinitely many points
+* for every open set <math>U</math> such that <math>x \in U</math>, the set <math>U \cap E</math> has infinitely many points
 * there exists a sequence <math>(x_n)_{n=1}^\infty</math> of distinct points in <math>E</math> (i.e. <math>x_n \in E</math> for all <math>n \geq 1</math> and <math>x_n \ne x_m</math> for all <math>n \ne m</math>) which converges to <math>x_0</math>
 * there exists a sequence <math>(x_n)_{n=1}^\infty</math> of points in <math>E</math>, none of which are equal to <math>x_0</math>, which converges to <math>x_0</math>