Comparison of Topologies

Definition

Comparison of Topologies

Consider a set $X$ with two topologies ${\mathcal{T}}_1$ and ${\mathcal{T}}_2$. If ${\mathcal{T}}_1\subset {\mathcal{T}}_2$, then ${\mathcal{T}}_1$ is weaker (smaller or coarser) than ${\mathcal{T}}_2$ and ${\mathcal{T}}_2$ is stronger (larger or finer) than ${\mathcal{T}}_1$.

Comparison of Topologies with Basis

Consider two topological spaces with bases $(X,\mathcal{T},\mathcal{B})$ and $(X,{\mathcal{T}}^{\prime },{\mathcal{B}}^{\prime })$. Then, $\mathcal{T}\subset {\mathcal{T}}^{\prime }⟺(\forall x\in X,\forall B\in \mathcal{B}\text{s.t.}x\in B),\exists B^{\prime }\subset {\mathcal{B}}^{\prime }\text{s.t.}{B}^{\prime }\subset B$