MTH435: Analysis & Topology - Exam 1 Vocab

Field Axioms

The collection of algebraic axioms governing addition and multiplication on a set, including commutativity, associativity, distributivity, and the existence of additive and multiplicative identities and inverses.

Order Axioms

The collection of axioms establishing an order relation ($<$) among real numbers that satisfies trichotomy, transitivity, and compatibility with addition and multiplication.

Theorem 1.1 ($\varepsilon$-Inequality)

States that given real numbers $a$ and $b$, if $a \le b + \varepsilon$ for every $\varepsilon > 0$, then $a \le b$.

Open Interval

A set defined by $(a, b) = \{x \in \mathbb{R} : a < x < b\}$.

Closed Interval

A set defined by $[a, b] = \{x \in \mathbb{R} : a \le x \le b\}$.

Inductive Set

A subset of real numbers containing $1$, with the property that $x + 1$ is in the set whenever $x$ is in the set.

Positive Integers ($\mathbb{Z}^+$)

The smallest inductive set of real numbers, formed by the intersection of all inductive subsets of $\mathbb{R}$.

Prime Number

An integer $n > 1$ whose only positive divisors are $1$ and $n$.

Unique Factorization Theorem

Every integer $n > 1$ can be factored into a product of prime numbers uniquely, up to the order of the prime factors.

Euclid's Lemma

If an integer $a$ divides a product $bc$ and $\gcd(a, b) = 1$, then $a$ divides $c$.

Rational Numbers ($\mathbb{Q}$)

The set of all numbers that can be expressed as a quotient of integers $\frac{a}{b}$, where $b \ne 0$.

Irrational Numbers

Real numbers that cannot be represented as the ratio of two integers (e.g., $\sqrt{2}$, $e$, $\pi$).

Upper Bound

A real number $b$ such that $x \le b$ for every element $x$ in a set $S$.

Maximum Element ($\max S$)

An upper bound of a set $S$ that also belongs to $S$.

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 $S$.

Completeness Axiom

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

Approximation Property of Supremum

If $b = \sup S$, then for every $a < b$, there exists an element $x \in S$ such that $a < x \le b$.

Archimedean Property

If $x > 0$ and $y$ is any real number, there exists a positive integer $n$ such that $nx > y$.

Triangle Inequality

The property stating that $|x + y| \le |x| + |y|$ for all real or complex numbers $x$ and $y$.

Cauchy-Schwarz Inequality

The inequality $\left(\sum_{k=1}^n a_k b_k\right)^2 \le \left(\sum_{k=1}^n a_k^2\right)\left(\sum_{k=1}^n b_k^2\right)$ for real sequences $a_k$ and $b_k$.

Extended Real Number System ($\mathbb{R}^*$)

The set $[-\infty, +\infty]$ formed by adjoining the symbols $+\infty$ and $-\infty$ to the real line $\mathbb{R}$.

Complex Number

An ordered pair of real numbers $(x_1, x_2)$, commonly expressed in the form $x_1 + i x_2$.

Imaginary Unit ($i$)

The complex number $(0, 1)$ satisfying the algebraic property $i^2 = -1$.

Principal Argument ($\arg(z)$)

The unique angle $\theta$ satisfying $-\pi < \theta \le \pi$ such that $x = |z|\cos(\theta)$ and $y = |z|\sin(\theta)$ for $z = x + iy \ne 0$.

Principal Logarithm ($\operatorname{Log}(z)$)

For a nonzero complex value $z$, defined by $\operatorname{Log}(z) = \ln|z| + i\arg(z)$.

Relation

Any collection of ordered pairs.

Domain

The set containing all the first components of ordered pairs within a relation.

Range

The set containing all the second components of ordered pairs within a relation.

Function

A relation in which no two distinct ordered pairs share the same first component.

Image of a Set

The set $F(S) = \{F(x) : x \in S\}$ containing all values mapped by a function $F$ from elements in $S$.

Injective Function

A mapping $F$ where distinct inputs yield distinct outputs, meaning $F(x) = F(y)$ implies $x = y$.

Inverse Function ($F^{-1}$)

The converse relation of a one-to-one mapping $F$, which maps every element in $\operatorname{range}(F)$ back to its unique pre-image in $\operatorname{domain}(F)$.

Composite Function ($G \circ F$)

A function defined by $(G \circ F)(x) = G(F(x))$, defined whenever the range of $F$ is contained within the domain of $G$.

Infinite Sequence

A function whose domain is the set of positive integers $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$.

Subsequence

A composite function $s \circ k$, where $s$ is an infinite sequence and $k$ is a strictly increasing function from $\mathbb{Z}^+$ into $\mathbb{Z}^+$.

Equinumerous Sets

Two sets $A$ and $B$ between which there exists a bijective (one-to-one and onto) mapping, denoted $A \sim B$.

Finite Set

A set that is equinumerous with $\emptyset$ or with the set of integers $\left\{1, 2, \dots, n\right\}$ for some positive integer $n$.

Countably Infinite Set

A set that can be placed in a one-to-one correspondence with the set of positive integers $\mathbb{Z}^+$, having cardinality $\aleph_0$.

Countable Set

A set that is either finite or countably infinite.

Uncountable Set

An infinite set that cannot be put into one-to-one correspondence with the set of positive integers (e.g., $\mathbb{R}$).

Union

The set of elements belonging to at least one set in a specified collection.

Intersection

The set of elements belonging to every set in a specified collection.

Complement ($B \setminus A$)

The relative set difference consisting of all elements belonging to set $B$ that do not belong to set $A$.

Disjoint Sets

Sets that share no common members, meaning their intersection is empty.

Euclidean Space $\mathbb{R}^n$

The vector space composed of all ordered $n$-tuples of real numbers $(x_1, x_2, \dots, x_n)$.

Inner Product

The scalar quantity defined for vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^n$ by $\mathbf{x} \cdot \mathbf{y} = \sum_{k=1}^n x_k y_k$.

Euclidean Norm

The geometric length of a vector $\mathbf{x} \in \mathbb{R}^n$, defined by $\left\|\mathbf{x}\right\| = \sqrt{\mathbf{x} \cdot \mathbf{x}} = \sqrt{\sum_{k=1}^n x_k^2}$.

Open $n$-ball

The set $B(\mathbf{a}; r) = \{\mathbf{x} \in \mathbb{R}^n : \left\|\mathbf{x} - \mathbf{a}\right\| < r\}$, consisting of all points located strictly within distance $r$ of center $\mathbf{a}$.

Interior Point

A point $\mathbf{a} \in S$ such that there exists an open ball $B(\mathbf{a}; r)$ completely contained within $S$.

Interior ($\operatorname{int} S$)

The collection of all interior points belonging to a set $S$.

Open Set

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

Component Interval

A maximal open subinterval of an open subset $S \subseteq \mathbb{R}$ that is not properly contained in any other open interval of $S$.

Representation Theorem for Open Sets in $\mathbb{R}^1$

States that every nonempty open set in $\mathbb{R}^1$ can be uniquely represented as a countable union of pairwise disjoint open component intervals.

Closed Set

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

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 $\mathbf{x}$.

Closure ($\overline{S}$)

The set of all adherent points of $S$, equivalent to the union of $S$ and its derived set ($\overline{S} = S \cup S'$).

Derived Set ($S'$)

The set consisting of all accumulation points of a given subset $S$.

Bounded Set

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

Bolzano-Weierstrass Theorem

States that every bounded, infinite subset of $\mathbb{R}^n$ contains at least one accumulation point in $\mathbb{R}^n$.

Cantor Intersection Theorem

States that for any nested sequence of nonempty, closed sets $Q_1 \supseteq Q_2 \supseteq \dots$ in $\mathbb{R}^n$ where $Q_1$ is bounded, the intersection $\bigcap_{k=1}^\infty Q_k$ is nonempty.

Open Covering

A family $\mathcal{F}$ of open sets whose total union contains a designated set $S$ ($S \subseteq \bigcup_{A \in \mathcal{F}} A$).

Lindelöf Covering Theorem

States that every open covering of a set $A \subseteq \mathbb{R}^n$ contains a countable subcollection that also covers $A$.

Heine-Borel Theorem

States that a subset of $\mathbb{R}^n$ is compact if and only if it is both closed and bounded.

Compact Set

A set $S$ with the property that every open covering of $S$ admits a finite subcovering.

Metric Space

A set $M$ equipped with a real-valued distance function $d$ satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality.

Relative Metric (Subspace)

The metric space $(S, d)$ resulting from restricting the distance function $d$ of a parent metric space $M$ to a subset $S \subseteq M$.

Boundary Point

A point $\mathbf{x} \in M$ where every open ball centered at $\mathbf{x}$ intersects both the set $S$ and its complement $M \setminus S$.

Boundary ($\partial S$)

The collection of all boundary points of $S$, given analytically by $\partial S = \overline{S} \cap \overline{M \setminus S}$.