Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. Let $A,B\in\mathcal F$ be events such that $A\cap B=\varnothing$, $\mathbb P(A)>0$, and $\mathbb P(B)>0$. Then $A$ and $B$ are not independent, meaning that $\mathbb P(A\cap B)\neq \mathbb P(A)\mathbb P(B)$.

Let $G$ be a group with identity element $e$, and let $H,K \le G$ be subgroups. Define a map \begin{align*} \alpha : (H \times K) \times G &\to G \\ ((h,k),g) &\mapsto h g k^{-1}. \end{align*} Then $\alpha$ is a group action of $H \times K$ on $G$, and for each $g \in G$, the orbit of $g$ under this action is exactly the double coset \begin{align*} HgK = \{h g k : h \in H,\ k \in K\}. \end{align*}

Let $G$ be a group with operation written multiplicatively, let $H \le G$ be a subgroup, and let $g \in G$. Define the left coset $gH := \{gh : h \in H\}$ and the right coset $Hg := \{hg : h \in H\}$. The maps \begin{align*} L_g: H &\to gH \\ h &\mapsto gh \end{align*} and \begin{align*} R_g: H &\to Hg \\ h &\mapsto hg \end{align*} are bijections. Consequently, if the relevant cardinalities are finite, then $|gH| = |H|$ and $|Hg| = |H|$.

Let $(G,\cdot)$ be a group with identity element $e$, and let $H \le G$ be a subgroup. Define a relation $\sim_R$ on $G$ by declaring that, for $x,y \in G$, \begin{align*} x \sim_R y \quad \Longleftrightarrow \quad yx^{-1} \in H. \end{align*} Then $\sim_R$ is an [equivalence relation](/page/Equivalence%20Relation) on $G$. Moreover, for each $x \in G$, the equivalence class of $x$ is the right coset \begin{align*} Hx := \{hx : h \in H\}. \end{align*}

Let $G$ be a group with multiplication written by juxtaposition, and let $e \in G$ denote its identity element. Let $H \le G$ be a subgroup, and let \begin{align*} G/H := \{gH : g \in G\} \end{align*} denote the set of left cosets of $H$ in $G$, where $gH := \{gh : h \in H\}$. Consider the left action \begin{align*} G \times G/H &\to G/H \\ (g, aH) &\mapsto (ga)H. \end{align*} If \begin{align*} G_H := \{g \in G : gH = H\} \end{align*} is the stabiliser of the base coset $H = eH$, then \begin{align*} G_H = H. \end{align*}

Let $(X,\tau)$ be a topological space, where $\tau$ is the collection of open subsets of $X$. A subset $F \subset X$ is closed if its complement $X \setminus F$ belongs to $\tau$. If $(F_i)_{i\in I}$ is any indexed family of closed subsets of $X$, then $\bigcap_{i\in I} F_i$ is closed. If $n \in \mathbb{N}$ and $F_1,\ldots,F_n$ are closed subsets of $X$, then $\bigcup_{i=1}^n F_i$ is closed.

Let $X$ be a set, and let $\mathcal B \subset \mathcal P(X)$ be a basis for a topology on $X$, meaning that $\bigcup_{B \in \mathcal B} B = X$ and whenever $B_1, B_2 \in \mathcal B$ and $x \in B_1 \cap B_2$, there exists $B_3 \in \mathcal B$ such that $x \in B_3 \subset B_1 \cap B_2$. Define \begin{align*} \tau_{\mathcal B} := \{U \subset X : \text{for every } x \in U, \text{ there exists } B \in \mathcal B \text{ such that } x \in B \subset U\}. \end{align*} Then $\tau_{\mathcal B}$ is a topology on $X$, and every element of $\tau_{\mathcal B}$ is a union of elements of $\mathcal B$.

Let $(X,\tau_X)$, $(Y,\tau_Y)$, and $(Z,\tau_Z)$ be topological spaces. Let $f: X \to Y$ and $g: Y \to Z$ be continuous maps, where continuity means that the preimage of every open set is open in the domain topology. Then the composition \begin{align*} g \circ f: X &\to Z \\ x &\mapsto g(f(x)) \end{align*} is continuous.

Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), and let $A \subset X$. Then the closure $\overline{A}$ is closed in $(X,\tau)$, satisfies $A \subset \overline{A}$, and has the following minimality property: for every subset $F \subset X$, if $F$ is closed in $(X,\tau)$ and $A \subset F$, then $\overline{A} \subset F$.

Let $X$ be a set equipped with the discrete topology $\tau_X = \mathcal{P}(X)$, let $(Y,\tau_Y)$ be a topological space, and let \begin{align*} f: X &\to Y \end{align*} be any map. Then $f: (X,\tau_X) \to (Y,\tau_Y)$ is continuous.