Definition:Empty Topological Space

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Let $\O$ denote the empty set.

The topological space $\struct {\O, \set \O}$ is called the empty topological space.

Also known as

The empty topological space is also known as the empty space if the context of topology is established.

Also see

  • Results about the empty topological space can be found here.

Topological Space may be Empty

Notwithstanding the result Empty Set Satisfies Topology Axioms, it is frequently stipulated in the literature that the class of topological spaces does not include the empty set.

This convention is sufficiently commonplace as to be often omitted in published texts, and taken for granted. When it is mentioned, it is usually given as an afterthought.

However, there exists a philosophical position that disallowing the empty topological space is unhelpful, and even harmful.

$\mathsf{Pr} \infty \mathsf{fWiki}$ adopts this philosophical position, allowing that the underlying set of a given topological space may indeed be empty.

Note, however, that many of the possible properties of a topological space are held vacuously by the empty space. This position having been taken, it is necessary in many cases to add a condition to a given general statement made about spaces specifically to exclude the empty space from the scope of that statement.