Editing Subbase (section)

===Alternative definition===

Less commonly, a slightly different definition of subbase is given which requires that the subbase <math>\mathcal{B}</math> cover <math>X.</math>{{sfn | Munkres | 2000 | pp=82}}  In this case, <math>X</math> is the union of all sets contained in <math>\mathcal{B}.</math> This means that there can be no confusion regarding the use of nullary intersections in the definition.

However, this definition is not always equivalent to the three definitions above. There exist topological spaces <math>(X, \tau)</math> with subcollections <math>\mathcal{B} \subseteq \tau</math> of the topology such that <math>\tau</math> is the smallest topology containing <math>\mathcal{B}</math>, yet <math>\mathcal{B}</math> does not cover <math>X</math>. For example, consider a topological space <math>(X,\tau)</math> with <math>\tau=\{\varnothing, \{p\}, X\}</math> and <math>\mathcal{B}=\{\{p\}\}</math> for some <math>p\in X.</math> Clearly, <math>\mathcal{B}</math> is a subbase of <math>\tau</math>, yet <math>\mathcal{B}</math> doesn't cover <math>X</math> as long as <math>X</math> has at least <math>2</math> elements. In practice, this is a rare occurrence. E.g. a subbase of a space that has at least two points and satisfies the [[T1 separation axiom|T<sub>1</sub> separation axiom]] must be a cover of that space.