Real Analysis Lecture Review - Lectures 1.1 to 4.2

Vocabulary practice flashcards generated from M365C Real Analysis lecture summary notes (Lectures 1.1 to 4.2).

Upper bound

An element $\beta$ for a set $E \subset X$ such that $\beta \ge x$ for all $x \in E$.

Supremum

The least upper bound

Infimum

The greatest lower bound

Neighborhood

The set $N_r(p) = \{q \in X : d(q, p) < r\}$ for $r > 0$ in a metric space $(X, d)$.

Limit point

A point $p \in X$ such that every neighborhood of $p$ contains a point of $E$ other than $p$, i.e., $N_r(p) \cap (E \setminus \{p\}) \neq \emptyset$ for all $r > 0$.

Isolated point

A point $p \in E$ that is not a limit point of $E$.

Interior point

A point $p \in X$ such that there exists $r > 0$ with $N_r(p) \subset E$.

Closure

The set $\overline{E} = E \cup E'$, where $E'$ is the set of all limit points of $E$.

Open set

A set $E$ in a metric space where every point of $E$ is an interior point of $E$.

Closed set

A set $E$ in a metric space that contains all of its limit points ($E' \subset E$).

Dense set

A subset $E \subset X$ such that every point of $X$ is in $E$ or is a limit point of $E$ (equivalently, $\overline{E} = X$).

epsilon characterization

B* = sup E <=> B* >= x for every x in E, and for every epsilon (€) > 0, there exists an x in E such that x > € - B*