Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $\tau_d$ be the topology on $X$ induced by $d$, and let $A \subset X$. For $x \in X$, the following are equivalent: 1. $x$ is a [limit point](/page/Limit%20Point) of $A$ with respect to $\tau_d$; that is, every $\tau_d$-[open set](/page/Open%20Set) $U \subset X$ with $x \in U$ satisfies $(U \setminus \{x\}) \cap A \ne \varnothing$. 2. For every $r > 0$, \begin{align*} (B(x,r) \setminus \{x\}) \cap A \ne \varnothing, \end{align*} where $B(x,r) := \{y \in X : d(x,y) < r\}$.

Let $(X,d)$ be a [metric space](/page/Metric%20Space), let $A \subset X$, and let $x \in X$. Then $x$ is a [limit point](/page/Limit%20Point) of $A$ if and only if there exists a sequence $(a_n)_{n=1}^{\infty}$ with $a_n \in A \setminus \{x\}$ for every $n \in \mathbb{N}$ such that $(a_n)$ converges to $x$ in the metric $d$, that is, \begin{align*} \lim_{n \to \infty} d(a_n,x) = 0. \end{align*}

Let $(X,\tau)$ be a $T_1$ [topological space](/page/Topological%20Space), let $A \subset X$, and let $x \in X$ be a [limit point](/page/Limit%20Point) of $A$, meaning that every neighbourhood of $x$ intersects $A \setminus \{x\}$. Then for every neighbourhood $N \subset X$ of $x$, the set $A \cap N$ is infinite.

Let $X$ be a set equipped with the discrete topology $\tau = \mathcal{P}(X)$. For every subset $A \subset X$, the derived set $A'$ of $A$ is empty: \begin{align*} A' = \varnothing. \end{align*}

Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), and let $A \subset X$. Define the derived set of $A$ by \begin{align*} A' := \{x \in X : \text{for every } U \in \tau \text{ with } x \in U,\; U \cap (A \setminus \{x\}) \neq \varnothing\}. \end{align*} Then $A$ is closed in $X$ if and only if $A' \subset A$.

Let $G$ be a group with operation written multiplicatively, and let $A \subset G$. For each $g \in G$, define the [conjugacy class](/page/Conjugacy%20Class) of $g$ in $G$ by \begin{align*} \operatorname{Cl}_G(g) := \{hgh^{-1} : h \in G\}. \end{align*} Then $A$ is conjugation-invariant, meaning that \begin{align*} \forall a \in A,\ \forall h \in G,\quad hah^{-1} \in A, \end{align*} if and only if $A$ is a union of conjugacy classes of $G$, meaning that there exists a subset $S \subset G$ such that \begin{align*} A = \bigcup_{s \in S} \operatorname{Cl}_G(s). \end{align*}

Let $(G,\cdot)$ be a group. Define a relation $\sim$ on $G$ by declaring that, for $g,k \in G$, \begin{align*} g \sim k \quad \Longleftrightarrow \quad \text{there exists } h \in G \text{ such that } k = hgh^{-1}. \end{align*} Then $\sim$ is an [equivalence relation](/page/Equivalence%20Relation) on $G$.

Let $G$ be a finite group, and let $\operatorname{Cl}(G)$ denote the set of conjugacy classes of $G$. Define the complex [vector space](/page/Vector%20Space) of class functions on $G$ by \begin{align*} \operatorname{CF}(G) := \{f: G \to \mathbb{C} \mid f(hgh^{-1}) = f(g) \text{ for all } g,h \in G\}. \end{align*} Then \begin{align*} \dim_{\mathbb{C}} \operatorname{CF}(G) = |\operatorname{Cl}(G)|. \end{align*}

Let $(G,\cdot)$ be a group with identity element $e$, and let $g \in G$. Define the centraliser of $g$ in $G$ by \begin{align*} C_G(g) := \{h \in G : h \cdot g = g \cdot h\}. \end{align*} Then $C_G(g) \le G$.

Let $G$ be a group, let $V$ be a finite-dimensional [vector space](/page/Vector%20Space) over $\mathbb{C}$, and let \begin{align*} \rho: G \to GL(V) \end{align*} be a group representation. Define the character of $\rho$ to be the map \begin{align*} \chi_\rho: G &\to \mathbb{C} \\ g &\mapsto \operatorname{tr}(\rho(g)). \end{align*} Then $\chi_\rho$ is a class function; equivalently, for every $g,h \in G$, \begin{align*} \chi_\rho(hgh^{-1}) = \chi_\rho(g). \end{align*}