Definition:Discrete Topology

From ProofWiki
Jump to navigation Jump to search

This page is about Discrete Topology. For other uses, see Definition:Discrete.

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 $S$:

$\tau := \set {H: H \subseteq S}$


Then $\tau$ is called the discrete topology on $S$ and $\struct {S, \tau} = \struct {S, \powerset S}$ the discrete space on $S$, or just a discrete space.


Finite Discrete Topology

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.


Infinite Discrete Topology

Let $S$ be an infinite set.

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


Also see

  • 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