Topology

Collection $\tau$ of subsets of $X$ is a topology if

1. $\varnothing, X \in \tau$
2. $\forall {U_{\alpha}}{\text{finite}} \subset \tau: \bigcap U{\alpha} \in \tau$
3. $\forall {U_{\alpha}}{\text{countable}} \subset \tau: \bigcup U{\alpha} \in \tau$

Each set in $\tau$ is called open-set. The $X$ is called a topological-space

Topology order

A topology $\tau_1$ is weaker than a topology $\tau_2$ if $\tau_1 \subset \tau_2$

• $\tau_{\text{weakest}} = {\varnothing, X}$
• $\tau_{\text{strongest}} = \mathcal{P}(X)$