Topological Space may be Empty

Philosophical Position
Notwithstanding the result Empty Set Satisfies Topology Axioms, it is usually 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.

follows this tradition, and (unless otherwise indicated) the assumption is that the underlying set of a given topological space is non-empty.

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

Many of the possible properties of a topological space are held vacuously by the empty space, and if this position is taken then 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.

Also see

 * Definition:Topological Space
 * Definition:Empty Set