# Category:Discrete Topology

This category contains results about Discrete Topology.

Definitions specific to this category can be found in Definitions/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**.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

### D

- Discrete Metrics (12 P)
- Discrete Space is Non-Meager (3 P)

### F

- Finite T1 Space is Discrete (3 P)

## Pages in category "Discrete Topology"

The following 72 pages are in this category, out of 72 total.

### C

- Convergence of Sequence in Discrete Space
- Convergence of Sequence in Discrete Space/Corollary
- Countable Discrete Space is Lindelöf
- Countable Discrete Space is not Weakly Countably Compact
- Countable Discrete Space is Second-Countable
- Countable Discrete Space is Separable
- Countable Discrete Space is Sigma-Compact

### D

- Discrete Space has Open Locally Finite Cover
- Discrete Space iff Diagonal Set on Product is Open
- Discrete Space is Compact iff Finite
- Discrete Space is Complete Metric Space
- Discrete Space is Extremally Disconnected
- Discrete Space is First-Countable
- Discrete Space is Fully Normal
- Discrete Space is Fully T4
- Discrete Space is Locally Connected
- Discrete Space is Locally Path-Connected
- Discrete Space is Non-Meager
- Discrete Space is not Dense-In-Itself
- Discrete Space is Paracompact
- Discrete Space is Scattered
- Discrete Space is Separable iff Countable
- Discrete Space is Strongly Locally Compact
- Discrete Space is Totally Disconnected
- Discrete Space is Zero Dimensional
- Discrete Space satisfies all Separation Properties
- Discrete Subspace of Fortissimo Space
- Discrete Topology is Finest Topology
- Discrete Topology is Metrizable
- Discrete Topology is Topology
- Discrete Uniformity generates Discrete Topology
- Double Pointed Discrete Real Number Space is not Lindelöf
- Double Pointed Discrete Real Number Space is Weakly Countably Compact

### E

### N

### P

- Particular Point Space less Particular Point is Discrete
- Particular Point Topology is Closed Extension Topology of Discrete Topology
- Partition of Singletons yields Discrete Topology
- Point in Discrete Space is Adherent Point
- Point in Discrete Space is Neighborhood
- Product of Countable Discrete Space with Sierpiński Space is Paracompact
- Properties of Discrete Topology

### S

### T

- Topological Space is Discrete iff All Points are Isolated
- Topologies induced by Usual Metric and Scaled Euclidean Metric on Positive Integers are Homeomorphic
- Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete
- Topology induced by Usual Metric on Positive Integers is Discrete
- Topology is Discrete iff All Singletons are Open
- Totally Disconnected and Locally Connected Space is Discrete