As part of my research of 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).