Definition:Indiscrete Topology
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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
- 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 $3.1.6$