Weak Topology Generated by Family of Functions

Definition

Consider a set $X$ and a family of functions $\{f_i:X\to X_i|i\in I\}$ from $X$ to each Topological Space $(X_i,{\mathcal{T}}_i)$. The weak topology generated by the family of functions $f_i$ is the Topology determined by the Subbasis of open sets $\{f_i^{-1}(U_i)|i\in I,U_i\in {\mathcal{T}}_i\}$.

Examples

Consider a metric space $(X,d)$, a point $a\in X$, and a function $f:X\to {\mathbb{R}}_0^{+},f(x)=d(a,x)$. Then the weak topology generated by the function $f$ is a Metric Topology determined by a Subbasis $f^{-1}([0,r))=B(a,r))$. It is an open ball with center $a$ and radius $r$.

Consider a set ${\mathbb{R}}^{\omega }$, and a family of projection functions $\{{\pi }_i:{\mathbb{R}}^{\omega }\to (\mathbb{R},\mathcal{U})\}$. The weak topology generated by the family of functions is a Product Topology determined by a Subbasis ${\pi }_i^{-1}(U)$ where $U\in \mathcal{U}$.

Facts

Consider a Topological Space $(X,\mathcal{T})$ and a family of continuous functions $\mathcal{F}=\{f_i:X\to Y_i\}$ with domain $X$. Then, the weak topology generated by the family of functions $\mathcal{F}$ is weaker than $\mathcal{T}$.