User:Ascii/ProofWiki Sampling Notes for Theorems/Topology

1. Indiscrete Topology is Coarsest Topology

   Let $T = \struct {S, \tau}$ be an indiscrete topological space.

   $\tau$ is the coarsest topology on $S$.

2. Discrete Topology is Finest Topology

   Let $T = \struct {S, \tau}$ be a discrete topological space.

   $\tau$ is the finest topology on $S$.

3. 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 $\mathbb T$: $\forall \tau_1, \tau_2 \in \mathbb T: \tau_1 \le \tau_2$ if and only if $\tau_1$ is coarser than $\tau_2$.

   Then $\le$ is a partial ordering on $\mathbb T$.

4. Topologies are not necessarily Comparable by Coarseness

   Let $S$ be a set with at least $2$ elements.

   There are always topologies on $S$ which are non-comparable.