Introducing a mathematical structure.

Question

Most of the times, introducing a new mathematical structure is done in the following path.

1. Start with a set/collection, name it as $X$. It is possible that $X$ already have a structure with it, namely the structure of topological space/manifold/vector space etc.
2. Define a structure on $X$, denote it by $\mathcal{A}$. So, a structure is a pair $(X,\mathcal{A})$.
3. Define what does it mean to say a substructure of $(X,\mathcal{A})$. Giving names to well-behaved substructures.
4. Define what are maps between two structures, say $(X,\mathcal{A})$ and $(Y,\mathcal{B})$. Giving names to well-behaved maps between two structures.
5. Define what does it mean to say two structures $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are "equivalent".
6. Construct new structures from old structures.

This step 6 differs drastically from one structure to another structure. Some ways to produce new structures from old structures are

1. Quotients.
2. Pullbacks.
3. Products (direct).
4. Sums (direct).
5. Limits (Injective/Projective).
6. ...
7. ...

For example, defining the notion of structure of a group on a set, we follow the same procedure. Defining a group, defining what does it mean to say a group morphism, what it means to say a subgroup (a normal subgroup), quotients of subgroup, (direct) sum, (direct) product of groups, (Injective/Projective) limit of (a collection) of groups and so on.

In this post, I want to collect this procedure for most of the structures introduced in undergraduate or beginning graduate courses in Mathematics.

Answer 1

Definition. Let $X$ be a set and let $d: X \times X \rightarrow [0, \infty)$ be a function satisfying the following requirements:

1. Positive definiteness: $d(x, x) = 0$ and whenever $d(x, y) = 0$, then $x = y$.
2. Symmetry: $d(x, y) = d(y,x)$
3. Triangle Inequality: $d(x, z) \leq d(x, y) + d(y, z)$.

Then the pair $(X, d)$ is called a metric space.

Substructures. A substructure in this case is just a subset of $X$ together with the restriction of $d$.

Maps. A map $f: X \rightarrow Y$ between metric spaces $(X, d)$ and $(Y, d')$ is continuous if it maps points that are close in $X$ to points that are close in $Y$. More concretely: the map $f$ is continuous if for any point $x \in X$ and any $\varepsilon > 0$, there exists $\delta >0$ such that whenever $d(x, y) \leq \delta$ we have $d'(f(x), f(y)) \leq \varepsilon$.

A bijection $f: X \rightarrow Y$ between metric spaces $(X, d)$ and $(Y, d')$ is called an isometry, if for all points $x, y \in X$ we have $d(x, y) = d(f(x), f(y))$. In this case, $X$ and $Y$ are said to be isometric, i.e., essentially equivalent as metric spaces.

Constructions. Let $(X, d)$ be a metric space and $f: Y \rightarrow X$ be an injection. Then the pullback metric $f^*d$ on $Y$ is given by $$(f^*d)(x, y) = d(f(x), f(y)).$$ This metric makes $f$ into an isometry onto its image.

Answer 2

Definition. Let $X$ be a set. Then a subset $\tau$ of the powerset of $X$ is a topology on $X$ if $\tau$ satisfies the following requirements.

1. $\emptyset, X \in \tau$.
2. Whenever $A_i \in \tau$ for $i\in I$, then $\cup_{i \in I}A_i \in \tau$.
3. Whenever $A, B \in \tau$, then $A \cap B \in \tau$.

In other words, $\tau$ contains the empty set, the whole set, is closed under arbitrary unions and is closed under finite intersection. In this case, the pair $(X, \tau)$ is called a topological space. To avoid mentioning the set $\tau$, the elements of $\tau$ are frequently called open sets.

Substructures. A subspace $X'$ of a topological space $(X, \tau)$ is a subset $X' \subset X$ equipped with the topology $\tau' = \{U \cap X' : U \in \tau \}$.

Maps. Let $X, Y$ be topological spaces. A map $f: X \rightarrow Y$ is called continuous if for any open subset $U \subset Y$ we have that $f^{-1}(U) \subset X$ is open. A bijection $f: X \rightarrow Y$ is called a homeomorphism if $f$ and $f^{-1}$ are both continuous. In this cases, the topological spaces are equivalent.

Constructions. Let $\{X_i\}_{i \in I}$ be a family of topological spaces. Then the product topology on $\prod_{i \in I}X_i$ is the smallest topology $\tau$ containing $\mathcal B = \{\prod_{i \in I} U_i : U_i \text{ is open and } U_i = X_i \text{ for all but finitely many } i \in I\}$. It is the coarsest topology such that all canonical projections are continuous.

For a family $\{X_i\}_{i \in I}$ of topological spaces, the disjoint union topology on $\bigsqcup_{i \in I}X_i$ is $\tau = \{\bigsqcup_{i \in I} U_i : U_i \subset X_i \text{ is open}\}$. It is the finest topology such that all canonical injections are continuous.

Answer 3

Definition. Let $G$ be a set and let $(\cdot):G \times G \rightarrow G$ be a binary operation, called multiplication. Then $(G, \cdot)$ is a group if

1. Asocciativity: $(g_1g_2)g_3 = g_1(g_2g_3)$ for all $g_1, g_2, g_3 \in G$.
2. Identity: There exists an element $e$ such that $eg = ge = g$ for all $g \in G$.
3. Inverses: For any $g \in G$ there exists an element $g^{-1} \in G$ such that $gg^{-1} = g^{-1}g = e$.

Substructures. A subgroup of a group $G$ is a subset of $G$ that is itself a group and the inclusion is a group homomorphism (see below).

Maps. A group homomorphism is a map $f: G \rightarrow H$ between groups such that for all $g, h \in G$ we have $f(gh) = f(g)f(h)$. A group isomorphism is a bijective group homomorphism.

Constructions. The product group of a family $\{G_i\}_{i \in I}$ of groups is obtained as follows. The underlying set is the product $\prod_{i \in I} G_i$ and the group operation is coordinatewise multiplication.

The direct sum of a family $\{G_i\}_{i \in I}$ of groups is very similar in construction to the product, but differs in one important detail: The underlying set is the subgroup $\bigoplus_{i \in I}G_i$ of $\prod_{i \in I}G_i$ such that all but finitely many components are equal to the identity.

A normal subgroup of a group is a subset $N$ of $G$ such that $gNg^{-1} = N$ for all $g \in G$. In this case, the quotient group $G/N = \{gN : g \in G\}$ has multiplication $gN \cdot hN = ghN$. The requirement that $N$ is normal is equivalent to the requirement that the multiplication on $G/N$ is well-defined.