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


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*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. 

*Define a structure on $X$, denote it by $\mathcal{A}$. So, a structure is a pair $(X,\mathcal{A})$. 

*Define what does it mean to say a substructure of $(X,\mathcal{A})$. Giving names to well-behaved substructures.

*Define what are maps between two structures, say $(X,\mathcal{A})$ and $(Y,\mathcal{B})$. Giving names to well-behaved maps between two structures.

*Define what does it mean to say two structures $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are "equivalent".

*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 


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*Quotients.

*Pullbacks.

*Products (direct).

*Sums (direct).

*Limits (Injective/Projective).

*...

*...


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. 
 A: 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


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*Asocciativity: $(g_1g_2)g_3 = g_1(g_2g_3)$ for all $g_1, g_2, g_3 \in G$.

*Identity: There exists an element $e$ such that $eg = ge = g$ for all $g \in G$.

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


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*Positive definiteness: $d(x, x) = 0$ and whenever $d(x, y) = 0$, then $x = y$.

*Symmetry: $d(x, y) = d(y,x)$

*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.
A: 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.


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*$\emptyset, X \in \tau$.

*Whenever $A_i \in \tau$ for $i\in I$, then $\cup_{i \in I}A_i \in \tau$.

*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.
