# Definition of a subbasis for a topology

Definition (Subbasis): A subbasis, $$\mathcal{S}$$, for a topology on $$X$$, is a collection of subsets of $$X$$, whose union equals $$X$$.

$$\mathcal{S} = \left\{S_{\alpha} \subset X \ \middle| \ \bigcup_{\alpha} S_{\alpha} = X \right\}$$

This is the definition of a subbasis that I've taken from Munkres: Topology - A First Course.

But by this definition if $$X$$ is a set, then a subbasis for a topology on $$X$$ could be $$\mathcal{S} = \{X\}$$ (as $$X \subset X$$ for any $$X$$). But then then the basis $$\mathcal{B}$$ generated by this subbasis, would be $$\mathcal{B} = \{X\}= \mathcal{S}$$, as the only finite intersection of elements of $$\mathcal{S}$$ is $$X\cap X = X$$.

But then we have for the topology $$\mathcal{T}$$ generated by this basis $$\mathcal{B}$$, $$\mathcal{T} = \{X\}$$ as the only possible union of basis elements is $$X$$. But $$\mathcal{T}$$ cannot be a topology as $$\emptyset \not\in \mathcal{T}$$, reaching a contradiction.

Have I made a mistake somewhere? If not then how can this definition of a subbasis be correct?

• You don't even need the union is $X$ condition, as the family of all finite intersections from $\mathcal{S}$ includes $X$ as the intersection of the empty family. Commented Jan 18, 2017 at 5:11

$X$ can be obtained as an empty union of elements of $\mathcal T$. An empty collection is still a collection.
• hi i have come across this question. What does your answer imply with respect to the question? I.e., can $\{X\}$ be a subbase of $X$? And where do we get the empty set from then to form a proper topology? Commented Mar 22, 2023 at 15:47