Definition:Indiscrete Topology

Definition

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

Let $\tau = \left\{{S, \varnothing}\right\}$.

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

A topological space $\left({S, \left\{{S, \varnothing}\right\}}\right)$ is known as an indiscrete space.

Also known as

• The trivial topology (on $S$)
• The nondiscrete topology (on $S$)

Also see

• Indiscrete Topology is Topology
• Indiscrete Topology is Coarsest Topology

• Results about indiscrete topologies can be found here.

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

• 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 3.2$: Topological Spaces: Example $3$
• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): $\S 1.1$
• 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology ... (previous) ... (next): $\text{II}: \ 4$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3.1$: Topological Spaces: Example $3.1.6$