User:Ascii/ProofWiki Sampling Notes for Theorems/Topology


 * 1) Indiscrete Topology is Coarsest Topology
 * Let $T = \left({S, \tau}\right)$ be an indiscrete topological space.
 * $\tau$ is the coarsest topology on $S$.
 * 1) Discrete Topology is Finest Topology
 * Let $T = \left({S, \tau}\right)$ be a discrete topological space.
 * $\tau$ is the finest topology on $S$.
 * 1) Coarseness Relation on Topologies is Partial Ordering
 * Let $X$ be a set and $\mathbb T$ be the set of all topologies on $X$.
 * And let $\le$ be the relation on $\mathbb T$: $\forall \tau_1, \tau_2 \in \mathbb T: \tau_1 \le \tau_2$ iff $\tau_1$ is coarser than $\tau_2$.
 * Then $\le$ is a partial ordering on $\mathbb T$.
 * 1) Topologies Not Always Comparable by Coarseness
 * Let $S$ be a set with at least $2$ elements.
 * There are always topologies on $S$ which are non-comparable.