Definition:Discrete Topology/Finite

Definition

Let $S \ne \O$ be a set.

Let $\tau = \powerset S$ be the power set of $S$.

That is, let $\tau$ be the set of all subsets of $S$:

$\tau := \set {H: H \subseteq S}$

Let $S$ be a finite set.

Then $\tau = \powerset S$ is a finite discrete topology, and $\struct {S, \tau} = \struct {S, \powerset S}$ is a finite discrete space.

Also see

• Definition:Infinite Discrete Topology
• Definition:Countable Discrete Topology
• Definition:Uncountable Discrete Topology

• Properties of Discrete Topology

• Results about discrete topologies can be found here.

Linguistic Note

Be careful with the word discrete.

A common homophone horror is to use the word discreet instead.

However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1$. Finite Discrete Topology