Definition:Indiscrete Topology

Not to be confused with Definition:Trivial Topological Space.

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 indiscrete topology on $S$ is 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 the indiscrete topology 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

• 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
• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Topological Spaces: Topologies