MTH435 - Analysis & Topology - Definitions for Exam 1

Component Interval

A maximal open subinterval of an open subset S ⊆ R that is not properly contained in any other open interval of S.

Open Set

A subset $S \subseteq \mathbb{R}^n$ where every element is an interior point, implying $S = \operatorname{int} S$.

Closed Set

A subset $S \subseteq \mathbb{R}^n$ whose complement $\mathbb{R}^n \setminus S$ is an open set.

Bounded Set

A set $S \subseteq \mathbb{R}^n$ that can be entirely enclosed inside some open ball $B(\mathbf{a}; r)$.

Monotone Convergence Theorem

states that a sequence of real numbers that is monotonic converges if, and only if, it is bounded, meaning an increasing sequence converges to its least upper bound (supremum) and a decreasing sequence converges to its greatest lower bound (infimum)

Supremum ($\sup S$)

The least upper bound of a set $S$ bounded above; the smallest real number serving as an upper bound for $S$.

Infimum ($\inf S$)

The greatest lower bound of a set $S$ bounded below; the largest real number serving as a lower bound for $S$.

Adherent Point

A point $\mathbf{x} \in \mathbb{R}^n$ such that every open ball $B(\mathbf{x}; r)$ contains at least one point in $S$.

Accumulation Point

A point $\mathbf{x} \in \mathbb{R}^n$ such that every open ball $B(\mathbf{x}; r)$ contains at least one point of $S$ distinct from $\mathbf{x}$.

Isolated Point

An element $\mathbf{x} \in S$ that is not an accumulation point of $S$, meaning there exists an open ball $B(\mathbf{x}; r)$ whose intersection with $S$ is only $\mathbf{x}$.

Completeness Axiom

The property stating that every nonempty set of real numbers that is bounded above possesses a supremum in $\mathbb{R}$.