📂Topology

Definition of Weak Topology

Definition ¹

1. Let $X$ be a set with two topologies $\mathscr{T}_{1}$, $\mathscr{T}_{2}$. If $\mathscr{T}_{1} \subset \mathscr{T}_{2}$, then $\mathscr{T}_{1}$ is said to be weaker than $\mathscr{T}_{2}$, and $\mathscr{T}_{2}$ is said to be stronger than $\mathscr{T}_{1}$.
2. Consider the set $\mathscr{F} := \left\{ f_{\alpha} : X \hookrightarrow X_{\alpha} , \alpha \in \mathscr{A} \right\}$ of injections from the set $X$ to the topological space $X_{\alpha}$.
   $$ \mathscr{S} := \left\{ f_{\alpha}^{-1} \left( O_{\alpha} \right) \subset X : \alpha \in \mathscr{A}, O_{\alpha} \text{ open in } X_{\alpha} \right\} $$ The topology on $X$ determined by having a set of subsets $\mathscr{S}$ as a subbasis is called the weak topology generated by the $f_{\alpha}$ on $X$.

Description

Weakest and Strongest Topologies

Regardless of the topological space, the trivial topology is the weakest, and the discrete topology is the strongest.

Strength of Topologies

That $\mathscr{T}_{1} \subset \mathscr{T}_{2}$ means the conditions that $\mathscr{T}_{1}$ has to satisfy to be a topology are weaker, also expressed as being coarser. Conversely, being stronger is also described as being finer.

Practical Appearance of Weak Topologies

Although defined as simply a collection of injections in the definition, what is often practically dealt with is a family of embeddings. In other words, $X$ is also a topological space, and the $f_{\alpha}$ are likely assumed to be injective continuous functions.