Definitions

Let \(V \subseteq M\). The complement of \(V\) in \(M\) is:

\[ M \setminus V := \{\, p \in M \mid p \notin V \,\}. \]

Let \((M, \mathcal{O}_M)\) and \((N, \mathcal{O}_N)\) be topological spaces and let \(f : M \to N\) be a map. Write \(\mathrm{preim}_f(V)\) for the preimage of \(V \subseteq N\).

The map \(f\) is continuous with respect to \(\mathcal{O}_M\) and \(\mathcal{O}_N\) if and only if:

\[ \forall\, V \in \mathcal{O}_N, \qquad \mathrm{preim}_f(V) \in \mathcal{O}_M. \]

In words: preimages of open sets are open.

Let \((M,\mathcal{O})\) be a \(d\)-dimensional topological manifold. An atlas of \((M,\mathcal{O})\) is a set of charts \[ \mathcal{A} = \{(U_{(\alpha)}, x_{(\alpha)}) \;\vert\; \alpha \in A\} \] whose domains cover \(M\): \[ \bigcup_{\alpha \in A} U_{(\alpha)} = M. \]

Let \((M,\mathcal{O})\) be a \(d\)-dimensional topological manifold. A chart of \((M,\mathcal{O})\) is a pair \((U,x)\) with \(U \in \mathcal{O}\) and \[ x : U \to x(U) \subseteq \mathbb{R}^d \] a homeomorphism onto an open subset of \(\mathbb{R}^d\). The map \(x\) is the chart map, and its components \[ x^1, \dots, x^d : U \to \mathbb{R}, \qquad x(p) = \big(x^1(p), \dots, x^d(p)\big) \] are the coordinate maps of the chart. The numbers \(x^1(p), \dots, x^d(p)\) are the coordinates of \(p\) with respect to \((U,x)\).

Let \((U,x)\) and \((V,y)\) be charts of a \(d\)-dimensional topological manifold with \(U \cap V \neq \emptyset\). The chart transition map from \((U,x)\) to \((V,y)\) is \[ y \circ x^{-1} : x(U \cap V) \to y(U \cap V). \]

Let \((M, \mathcal{O})\) be a topological space. A separation of \(M\) is a pair of open sets \(U, V \in \mathcal{O}\) with

\[ U \cap V = \emptyset, \qquad U \neq \emptyset, \qquad V \neq \emptyset, \qquad U \cup V = M. \]

\(M\) is disconnected if a separation of it exists, and connected if none does.

Let \(U_p\) denote an open neighbourhood containing the point \(p\). A topological space \((M,\mathcal{O})\) is called a \(d\)-dimensional topological manifold if \[ \forall p \in M \;\; \exists U_p \in \mathcal{O} \;\; : \;\; \exists x : U_p \to x(U_p) \subseteq \mathbb{R}^d \] such that

(i) \(x\) is invertible, \(x^{-1} : x(U_p) \to U_p\);
(ii) \(x\) is continuous;
(iii) \(x^{-1}\) is continuous;
(iv) \(x(U_p)\) is open with respect to the standard topology \(\mathcal{O}_s\) on \(\mathbb{R}^d\).

Continuity is asked of \(U_p\) and \(x(U_p)\) carrying the subset topologies they inherit from \(\mathcal{O}\) and from \(\mathcal{O}_s\).

The chaotic topology on \(M\), elsewhere called the indiscrete or trivial topology, is \(\mathcal{O}_{\text{chaotic}} := \{\emptyset, M\}\), the smallest collection the conditions on a topology permit.

The discrete topology on \(M\) is \(\mathcal{O}_{\text{discrete}} := \mathcal{P}(M)\), in which every subset is open.

Let \((M, \mathcal{O})\) be a topological space. A family \(\mathcal{C} \subseteq \mathcal{O}\) of open sets is an open cover of \(M\) if every point of \(M\) lies in at least one member of \(\mathcal{C}\), that is \(\bigcup \mathcal{C} = M\).

A subcover is a subfamily \(\mathcal{C}' \subseteq \mathcal{C}\) that is still a cover. It is a finite subcover when \(\mathcal{C}'\) has finitely many members.

\(M\) is compact if every open cover of \(M\) has a finite subcover.

Let \(f : M \to N\) be a bijective map between topological spaces \((M, \mathcal{O}_M)\) and \((N, \mathcal{O}_N)\). Then \(f\) is a homeomorphism if both \(f\) and its inverse \(f^{-1}\) are continuous.

Let \(f : M \to N\) be a map, \(M\) the domain and \(N\) the target.

\(f\) is injective if and only if \(f(p_1) = f(p_2) \iff p_1 = p_2\) for all \(p_1, p_2 \in M\).

\(f\) is surjective if and only if for every \(n \in N\) there is a \(p \in M\) with \(f(p) = n\).

\(f\) is bijective if and only if it is both.

Let \((M, \mathcal{O})\) be a topological space.

A subset \(U \subseteq M\) is open if and only if \(U \in \mathcal{O}\).

A subset \(V \subseteq M\) is closed if and only if its complement \(M \setminus V\) is in \(\mathcal{O}\).

The powerset \(\mathcal{P}(S)\) of a set \(S\) is the set of all subsets of \(S\).

Let \(f : M \to N\) be a map, \(M\) the domain and \(N\) the target, and let \(V \subseteq N\). The preimage of \(V\) under \(f\) is:

\[ \mathrm{preim}_f(V) := \{\, p \in M \mid f(p) \in V \,\}. \]

For \(p \in \mathbb{R}^d\) and \(r \in \mathbb{R}^+\), the soft ball of radius \(r\) about \(p\) is:

\[ B_r(p) := \Big\{\, (q_1, \dots, q_d) \in \mathbb{R}^d \;\Big|\; \textstyle\sum_{i=1}^{d} (q_i - p_i)^2 < r^2 \,\Big\}. \]

The standard topology on \(\mathbb{R}^d\) selects from the powerset of \(\mathbb{R}^d\):

\[ \mathcal{O}_s := \big\{\, U \in \mathcal{P}(\mathbb{R}^d) \;\big|\; \forall\, p \in U \;\; \exists\, r \in \mathbb{R}^+ : B_r(p) \subseteq U \,\big\}. \]

Let \((M, \mathcal{O}_M)\) be a topological space and let \(S \subseteq M\). The subset topology on \(S\) is:

\[ \mathcal{O}\vert_S := \{\, U \cap S \mid U \in \mathcal{O}_M \,\}. \]

Let \(M\) be a set and \(\mathcal{O}\) a topology on \(M\). The double \((M, \mathcal{O})\) is called a topological space.

Let \(M\) be a set. A topology \(\mathcal{O}\) on \(M\) is a subset of the powerset, \(\mathcal{O} \subseteq \mathcal{P}(M)\), satisfying:

1. \(\emptyset \in \mathcal{O}\) and \(M \in \mathcal{O}\);
2. given \(U, V \in \mathcal{O}\), then \(U \cap V \in \mathcal{O}\);
3. let \(A\) be an arbitrary index set; given \(U_\alpha \in \mathcal{O}\), then \(\bigcup_{\alpha \in A} U_\alpha \in \mathcal{O}\).