nLab finer topology

Definition

Definition

(finer/coarser topologies)

Let $X$ be a set, and let $\tau_1, \tau_2 \subset P(X)$ be two topologies on $X$, hence two choices of open subsets for $X$, making it a topological space. If

$$\tau_1 \subset \tau_2$$

hence if every open subset of $X$ with respect to $\tau_1$ is also regarded as open by $\tau_2$, then one says that

• the topology $\tau_2$ is finer than the topology $\tau_1$

• the topology $\tau_1$ is coarser than the topology $\tau_2$.

Examples

Example

(discrete and co-discrete topology)

Let $X$ be any set. Then there are always the following two extreme possibilities of equipping $X$ with a topology $\tau \subset P(X)$, and hence making it a topological space:

1. $\tau \coloneqq P(X)$ the set of all subsets;

   this is called the discrete topology on $X$, it is the finest topology (def. ) on $X$,

   we write $Disc(X)$ for the resulting topological space;

2. $\tau \coloneqq \{ \emptyset, X \}$ the set contaning only the empty subset of $X$ and all of $X$ itself;

   this is called the codiscrete topology on $X$, it is the coarsest topology (def. ) on $X$

   we write $CoDisc(X)$ for the resulting topological space.

References

• Wikipedia, Comparison of topologies