This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Notation

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Generalities:

• $\Rightarrow \quad$ “therefore”;
• $\Leftrightarrow \quad$ “if and only if”;
• $\forall \quad$ “for any” or “for all”;
• $\exists \quad$ “there exists” or “for some”;
• $A:=$ or $=:A \quad$ “$A$ is defined as”;
• $A \setminus B:= \{x\in A:\ x\not\in B\}$;
• $A^n:=\{(x_1,...,x_n):x_i\in A\} \quad$ the $n$th power of set $A$;
• $f:X\to Y \quad$ a function from set $X$ to set $Y$;
• $i_X:A \hookrightarrow X \quad$ the inclusion function of subset $A\subset X$ of set $X$ into $X$;
• ${\rm Id}_X:X\to X \quad$ the identity function on set $X$;
• $f:x\mapsto y \quad$ function $f$ takes $x\in X$ to $y \in Y$, i.e., $f(x)=y$;
• $\#A \quad$ the cardinality of set $A$;
• $f(A) :=\{y\in Y:y=f(x),x\in A\} \quad$ the image of subset $A\subset X$ under $f:X\to Y$;
• $\operatorname{Im} f :=f(X) \quad$ the (total) image of $f:X\to Y$;
• $\operatorname{Graph} f :=\{(x,y):y=f(x)\} \subset X\times Y \quad$ the graph of function $f:X\to Y$;
• $2^X:=\{A\subset X\} \quad$ the power set of set $X$.

Basic topology:

• $\operatorname{Cl}(A) \quad$ the closure of subset $A\subset X$ in $X$;
• $\operatorname{Int}(A) \quad$ the interior of subset $A\subset X$ in $X$;
• $\operatorname{Fr}(A) \quad$ the frontier of subset $A\subset X$ in $X$;
• $B(a,\delta)= \{u\in {\bf R}^n : ||u-a|| < \delta \} \quad$ the open ball centered at $a\in {\bf R}^n$ of radius $\delta$;
• $\bar{B}(a,\delta)= \{u\in {\bf R}^n : ||u-a|| \le \delta \} \quad$ the closed ball centered at $a\in {\bf R}^n$ of radius $\delta$;
• $\dot{\sigma} \quad$ the interior/inside of cell $\sigma$;
• $F(X,Y) \quad$ the set of all functions $f:X\to Y$;
• $C(X,Y) \quad$ the set of all maps $f:X\to Y$;
• $C(X):=C(X,{\bf R})$, or $C(X,R)$ for some ring$R$.

Sets:

• ${\bf R} \quad$ the real numbers;
• ${\bf C} \quad$ the complex numbers;
• ${\bf Q} \quad$ the rational numbers;
• ${\bf Z} \quad$ the integers;
• ${\bf Z}_n:=\{0,1,...,n-1\} \quad$ the integers modulo $n$;
• ${\bf R}^n:=\{(x_1,...,x_n):x_i\in {\bf R}\} \quad$ the $n$-dimensional Euclidean space;
• ${\mathbb R}^n \quad$ the standard cubical complex representation -- with unit cubes -- of ${\bf R}^n$;
• ${\bf R}^n_+:=\{(x_1,...,x_n):x_i\in {\bf R},x_1 \ge 0\} \quad$ the positive half-space of ${\bf R}^n$;
• ${\bf B}^n := \{u\in {\bf R}^n: ||u|| \le 1\} \quad$ the closed unit ball in ${\bf R}^n$;
• ${\bf S}^{1} \quad$ the circle;
• ${\bf S}^{n-1} := \{u\in {\bf R}^n: ||u|| = 1\} \quad$ the unit sphere in ${\bf R}^n$;
• ${\bf I}:=[0,1] \quad$ the closed unit interval;
• ${\bf I}^n \quad$ the $n$-cube;
• ${\bf T}={\bf T}^2 \quad$ the torus;
• ${\bf T}^n := \ {\bf S}^1 \times {\bf S}^1 .. \times {\bf S}^1 \quad$ the $n$-torus;
• ${\bf M}={\bf M}^2 \quad$ the Mobius band;
• ${\bf P}={\bf P}^2 \quad$ the projective plane;
• ${\bf K}={\bf K}^2 \quad$ the Klein bottle.

Algebra:

• $\langle x,y \rangle \quad$ the inner product;
• $<A|B> \quad$ the subgroup of group $G$ generated by the subset $A \subset G$ with condition $B$ (the span in the case of vector spaces);
• $<a>:=\{na:n\in {\bf Z}\} \quad$ the cyclic subgroup of group $G$ generated by element $a\in G$;
• $x \sim y \quad$ equivalence of elements with respect to equivalence relation $\sim$;
• $[a]:=\{x\in A: x\sim a\} \quad$ the equivalence class of $a\in A$ with respect to equivalence relation $\sim$;
• $A /_{\sim } := \{[a]:a\in A\} \quad$ the quotient of set $A$ with respect to equivalence relation $\sim$;
• $[f]:A /_{\sim } \to B / _{\sim} \quad$ the quotient function of function $f:A \to B$ with respect to these two equivalence relations, i.e., $[f]([a]):=[f(a)]$;
• $A\times B := \{(a,b):a\in A,b\in B\} \quad$ the product of sets $A,B$;
• $G\oplus H := \{(a,b):a\in G,b\in H\} \quad$ the direct sum of abelian groups (or vector spaces) $G,H$;
• $g\oplus h :G\oplus H \to G'\oplus H' \quad$ the direct sum of homomorphisms (or linear operators) $g:G\to G',h:H \to H'$, i.e., $(g\oplus h) (a,b) := (g(a),h(b))$;
• $\mathcal{S}_n \quad$ the group of permutations of $n$ elements;
• $\mathcal{A}_n \quad$ the group of even permutations of $n$ elements;
• commutative diagrams and a non-commutative diagram:

$$\newcommand{\ra}[1]{\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \newcommand{\la}[1]{\!\!\!\!\!\!\!\xleftarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\ua}[1]{\left\uparrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} \begin{array}{ccccc} A & \ra{} & B \\ & \searrow & \da{} \\ & & C \end{array} \quad \quad \begin{array}{ccccc} A & \ra{} & B \\ \da{}& & \da{}\\ D& \ra{}& C \end{array} \quad \quad \begin{array}{ccccc} A & \ra{} & B \\ \da{}& \ne & \da{}\\ D&\ra{} & C \end{array}$$

Homology: $X,Y$ cell complexes, $M,N$ chain complexes

• $C_n=C_n(X) \quad$ the group of $n$-chains of cell complex $X$;
• $\partial _n:C_n(X)\to C_{n-1}(X) \quad$ the $n$th boundary operator of $X$;
• $C(X):=\{C_n(X):n=0,1,2,...\}$, or $C(X):=\oplus _n C_n(X) \quad$ the (total) chain group of $X$;
• $\partial =\{\partial _n:C_n(X)\to C_{n-1}(X)\}:C(X) \to C(X) \quad$ the (total) boundary operator of $X$;
• $C(X)=\{C(X),\partial\} \quad$ the chain complex of $X$;
• $f_n:C_n(X)\to C_n(Y) \quad$ the $n$th chain map of map $f:X\to Y$;
• $f_{\Delta}:=\{f_n:n=0,1,2...\}:C(X)\to C(Y)$, or $f_{\Delta}:=\oplus _n f_n \quad$ the (total) chain map of map $f:X\to Y$;
• $Z_n=Z_n(X):=\ker \partial _n \quad$ the group of $n$-cycles of $X$, or
• $Z_n(M):=\ker \partial _n \quad$ the group of $n$-cycles of $M$;
• $B_n=B_n(X):=\operatorname{Im} \partial _{n+1} \quad$ the group of $n$-boundaries of $X$, or
• $B_n(M):=\operatorname{Im} \partial _{n+1} \quad$ the group of $n$-boundaries of $M$;
• $H_n=H_n(X):=Z_n(X) / B_n(X) \quad$ the $n$th homology group of $X$, or
• $H_n(M):=Z_n(M) / B_n(M) \quad$ the $n$th homology group of $M$;
• $H(X):=\{H_n(X):n=0,1,2,...\}$, or $H(X):=\oplus _n H_n(X) \quad$ the (total) homology group of $X$, or
• $H(M) \quad$ the (total) homology group of $M$;
• $\beta_n(X) := \dim H_n(X) \quad$ the $n$th Betti number of $X$, or
• $\beta_n (M) := \dim H_n(X) \quad$ the $n$th Betti number of $M$;
• $[f_n]:=[f_{\Delta}]:H_n(X)\to H_n(Y) \quad$ the $n$th homology map of map $f:X\to Y$, or
• $[g_n]:H_n(M)\to H_n(N) \quad$ the $n$th homology map of the chain map $g=\{g_n:n=0,1,2...:M_n\to N_n\}$;
• $f_*:=\{[f_n]:n=0,1,2...\}:H(X)\to H(Y)$, or $f_*=\oplus _n [f_n] \quad$ the (total) homology map of map $f:X\to Y$, or
• $g_*:=\{[g_n]:n=0,1,2...\}:H(M)\to H(N)\quad$ the (total) homology map of the chain map $g:M\to N$;
• Same for $C_n(X,A;R),H_n(X,A;R) \quad$ the $n$th chain group and the $n$th homology group of the pair $(X,A)$ over ring $R$.

Other algebraic topology:

• $X \sqcup Y \quad$ the disjoint union of $X,Y$;
• $X \vee Y := \left(X \sqcup Y \right) /\{p\} \quad$ the one-point union of spaces $X,Y$;
• $\Sigma X:=[0,1] \times X / _{\{(0,x)\sim (0,y),(1,x)\sim (1,y)\}} \quad$ the suspension of $X$;
• $\chi(X) \quad$ the Euler characteristic of $X$;
• $[X,Y] \quad$ the set of all homotopy classes of maps $X\to Y$;
• $\pi_1(X) \quad$ the fundamental group of $X$;
• $|K| \quad$ the realization of complex $K$;
• ${\rm St}_A(K) \quad$ the star of vertex $A$ in complex $K$;
• $T_A(K) \quad$ the tangent space of vertex $A$ in complex $K$;
• $T(K) \quad$ the tangent bundle of complex $K$;
• $K^{(n)} \quad$ the $n$th skeleton of complex $K$;
• $A_0A_1 ... A_n \quad$ the $n$-simplex with vertices $A_0,A_1, ..., A_n$;
• $G \cong H \quad$ isomorphism;
• $X \approx Y \quad$ homeomorphism;
• $X \simeq Y \quad$ homotopy equivalence;
• $f \simeq g:X\to Y \quad$ homotopy of maps.

Notation not used:

• $3\frac{1}{3}$
• $10\%$
• $90^o$
• $\vec{v}$
• $8\div 4$
• $\log x$
• $f\circ g$
• $3\times 5$

Formatting:

• In bold:
• Theorem., sometimes with its name included, in caps, as in
• Theorem (Simplicial Approximation Theorem).,
• Definition.,
• Example.,
• Exercise.,
• Proof..
• Proofs end with $\blacksquare$.
• Examples end with $\square$.
• What is being defined is in italics.