Indiscrete space

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Definition

A topological space is termed an indiscrete space if it satisfies the following equivalent conditions:

  1. It has an empty subbasis.
  2. It has a basis comprising only the whole space.
  3. The only open subsets are the whole space and the empty subset.
  4. The only closed subsets are the whole space and the empty subset.
  5. The space is either an empty space or its Kolmogorov quotient is a one-point space.

For any set, there is a unique topology on it making it an indiscrete space. This topology is called the indiscrete topology or the trivial topology. It is the coarsest possible topology on the set.

In some conventions, empty spaces are considered indiscrete. In other conventions, we exclude empty spaces from consideration.

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
ultraconnected space |FULL LIST, MORE INFO
normal space |FULL LIST, MORE INFO
regular space |FULL LIST, MORE INFO
preregular space |FULL LIST, MORE INFO
symmetric space |FULL LIST, MORE INFO
contractible space