What topological properties are trivially/vacuously satisfied by any indiscrete space

compactnessgeneral-topologyseparation-axioms

A space $X$ is indiscrete provided its topology is $\{\emptyset,X\}$. With such a restrictive topology, such spaces must be examples/counterexamples for many other topological properties. Then my question is:

Question: What topological properties are trivially/vacuously satisfied by any indiscrete space?

Best Answer

Separation properties

Any indiscrete space is perfectly normal (disjoint closed sets can be separated by a continuous real-valued function) vacuously since there don't exist disjoint closed sets. But on the other hand, the only T0 indiscrete spaces are the empty set and the singleton.

Metrizability

Only the empty and singleton indiscrete spaces are metrizable, but every indiscrete space is compatible with the pseudometric $d(x,y)=0$ for all $x,y$.

Covering properties

Any indiscrete space is compact since its only open cover is finite to begin with ($\{X\}$).

Topological size

Every basis for an indiscrete space is finite ($\Rightarrow$ countable), so it is second-countable and therefore separable.

Connectedness

$X$ is the only nonempty clopen set, so indiscrete spaces are connected. They are also:

  • Strongly connected, that is, the only continuous functions $f:X\to\mathbb R$ are constant.
  • Hyperconnected, that is, all nonempty open sets intersect
  • Ultraconnected, that is, all nonempty closed sets intersect
  • Path connected since all maps [from $\mathbb R$] to $X$ are continuous. This strengthens to arc connected if the space has the cardinality of the reals or greater (as arc connected requires injectivity).
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