As part of my research of http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf I came across a myriad of different topologies that could be put on various spaces associated with a space $X$. This piece is my attempt to collate and organise them. All are examples of initial topologies. An initial topology is determined by a family of functions.

Initial Topology (Belton 1.24)

Let $X$ be a set and $F$ be collection of functions on $X$, such that $f : X \rightarrow Y_f$ , where $(Y_f,\tau_f)$ is a topological space, for all $f \in F$. The initial topology generated by $F$, denoted by $\tau_F$ , is the coarsest topology such that each function $f\in F$ is continuous.  It is clear that $\tau_F$ is the intersection of all topologies on $X$ that contain

$\bigcup_{f\in F} f^{-1}(\tau_f)=\{f^{-1}(U):f\in F,\,U\in \tau_f\}$

Product Topology (Belton 3.20)

Let $\{(X_a, \tau_a) : a \in A\}$be a collection of topological spaces. Their topological product is $(X, T)$, where ($\sum$ is Cartesian product)

$X =\sum_{a\in A} X_a:=\{(x_a)_{a\in A}: x_a\in X_a\,,\,\forall\,a\in A\}$

is the Cartesian product of the sets $X_a$ and $\tau$ is the initial topology generated by the projection maps, for $b\in A$ :

$\pi_b:X\rightarrow X_b;(x_a)_{a\in A}\mapsto x_b$

Strong Operator Topology (Belton 2.22)

Let $X$ and $Y$ be normed spaces; the initial topology on the bounded linear operators $X\rightarrow Y$, $B(X, Y)$ generated by the family of maps $\{T \mapsto Tx : x \in X\}$ (where $Y$ is equipped with its norm topology) is called the strong operator topology.

Weak Topology (Belton 3.4)

Any normed space $X$ gains a natural topology from its dual space, its weak topology. This is the initial topology generated by $X^*$, i.e., the coarsest topology to make each map $\varphi\in X^*$ continuous. The weak topology on $X$ is denoted by $\sigma(X,X^*)$.

Weak* Topology (Belton 3.19)

Let $X$ be a normed vector space. The weak* topology on $X^*$ is the initial topology generated by the maps $\hat{x}: X^* \rightarrow \mathbb{F}; \varphi \mapsto \varphi(x),\,\, (x \in X)$,i.e., the coarsest topology to make these maps continuous. The weak* topology on $X^*$ is denoted by $\sigma(X^*,X)$.