Category:Definitions/Discrete Topology

This category contains definitions related to Discrete Topology.
Related results can be found in Category:Discrete Topology.

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.