Catalog of Definitions from Real Analysis.

Definition 12. Limit Point
§4.3
A point x∈Rn is a limit point of a set A⊆Rnif ∃(ak)⊆A such that lim⁡k→∞ak=x.\text{A point } \mathbf{x} \in \mathbb{R}^n \text{ is a limit point of a set } A \subseteq \mathbb{R}^n \\ \text{if } \exists (\mathbf{a}_k) \subseteq A \text{ such that } \lim_{k \to \infty} \mathbf{a}_k = \mathbf{x}.
Definition 13. Closed
§4.3
A set A⊆Rn is closed if it containsall of its limit points‾.In other words, Closed means {limit pts of A}⊆A.Note for any a∈A, we may consider the constantsequence ak≡a ∀k to see a∈{limit pts of A}.That is, {limit pts of A}⊇A always holds.So A is closed iff A={limit pts of A}, and to showA is closed we only need to show all limit points are in A.\text{A set } A \subseteq \mathbb{R}^n \text{ is closed if it contains} \\ \text{all of its \underline{\href{/notes/real-analysis/definitions#definition-12}{\text{limit points}}}.} \\[1em] \text{In other words,} \text{ Closed means } \{\text{limit pts of } A\} \subseteq A. \\[1em] \text{Note for any } \mathbf{a} \in A, \text{ we may consider the constant} \\ \text{sequence } \mathbf{a}_k \equiv \mathbf{a}\ \forall k \text{ to see } \mathbf{a} \in \{\text{limit pts of } A\}. \\ \text{That is, } \{\text{limit pts of } A\} \supseteq A \text{ always holds.} \\ \text{So } A \text{ is closed iff } A = \{\text{limit pts of } A\}, \text{ and to show} \\ A \text{ is closed we only need to show all limit points are in } A.
Definition 14. Closure
§4.3
Let A⊆Rn. The closure of A is the set A‾consisting of all of its limit points‾.Note A closed‾  ⟺  A‾⊆A.\text{Let } A \subseteq \mathbb{R}^n. \text{ The closure of } A \text{ is the set } \overline{A} \\ \text{consisting of all of its \underline{\href{/notes/real-analysis/definitions#definition-12}{\text{limit points}}}.} \\[1em] \text{Note } A \text{ \underline{\href{/notes/real-analysis/definitions#definition-13}{\text{closed}}}} \iff \overline{A} \subseteq A.
Definition 15. Dense
§4.3
A set A is dense in B if A⊆B⊆A‾.e.g. Q is dense in R, since Q‾=R.\text{A set } A \text{ is dense in } B \text{ if } A \subseteq B \subseteq \overline{A}. \\[1em] \text{e.g. } \mathbb{Q} \text{ is dense in } \mathbb{R}, \text{ since } \overline{\mathbb{Q}} = \mathbb{R}.
Definition 16. Ball
§4.3
The ball about a∈Rn of radius r is the setBr(a):={x∈Rn∣∥x−a∥<r}.\text{The ball about } \mathbf{a} \in \mathbb{R}^n \text{ of radius } r \text{ is the set} \\ B_r(\mathbf{a}) := \{ \mathbf{x} \in \mathbb{R}^n \mid \lVert \mathbf{x} - \mathbf{a} \rVert < r \}.
Definition 17. Open
§4.3
A set U⊆Rn is open if∀a∈U, ∃r=r(a)>0, Br(a)‾⊆U.\text{A set } U \subseteq \mathbb{R}^n \text{ is open if} \\ \forall \mathbf{a} \in U,\ \exists r = r(\mathbf{a}) > 0,\ \underline{\href{/notes/real-analysis/definitions#definition-16}{\text{$B_r(\mathbf{a})$}}} \subseteq U.
Definition 18. Interior
§4.3
The interior of a set X⊆Rn is the largestopen‾ set contained in X, and is denoted int⁡(X).\text{The interior of a set } X \subseteq \mathbb{R}^n \text{ is the largest} \\ \text{\underline{\href{/notes/real-analysis/definitions#definition-17}{\text{open}}} set contained in } X, \text{ and is denoted } \operatorname{int}(X).
Black and white dog lying belly-up on a bed