# Definition:Indiscrete Topology

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## Definition

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

Let $\tau = \set {S, \O}$.

Then $\tau$ is called the **indiscrete topology** on $S$.

A topological space $\struct {S, \set {S, \O} }$ is known as an **indiscrete space**.

## Also known as

- The
**trivial topology**(on $S$) - The
**nondiscrete topology**(on $S$)

## Also see

- Results about
**indiscrete topologies**can be found**here**.

*Not to be confused with Definition:Trivial Topological Space.*

## Linguistic Note

Be careful how you spell **indiscrete**. A common homophone horror is to refer to this as the **indiscreet topology**.

However, **indiscreet** means **incautious** or **tactless**. It's how you describe somebody who cannot keep a secret.

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 2$: Topological Spaces: Example $3$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Example $3.1.6$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $4$. Indiscrete Topology