What topological structures have exactly one base?

Question

for homework

What topological structures have exactly one base?

I think that:

$\{\emptyset,X\}$ his basis is $\{\emptyset,X\}$

is this ok? I am not sure, could please provide me another example and why?

Answer 1

Think about $\mathbb R$ with the set consisting of all intervals $(-1/n,\ 1/n)$ for natural $n$.

Remark: You can show that the topologies on a set $X$ with only one base are exactly the families $\mathcal S$ of subsets of $X$ such that $\{\emptyset,X\}\subseteq\mathcal S$ and every subfamily $\mathcal T\subseteq\mathcal S$ has a greatest element, i.e. a set $M\in\mathcal T$ which contains all other $T\in\mathcal T$.

Answer 2

For any finite sequence $\emptyset \subseteq X_1 \subseteq X_2 \subseteq \dots \subseteq X_n \subseteq X$ the topology on $X$ consisting of those sets provides another example.

Task: are there further examples?

Answer 3

The definition that I learned, ages ago, is that a base for a topology $\mathcal T$ is a family $\mathcal B\subseteq\mathcal T$ such that every member of $\mathcal T$ is the union of a subfamily of $\mathcal B$. ("Every open set is a union of some basic open sets.") With this definition, $\mathcal T$ is a base for itself, and so is $\mathcal T-\{\varnothing\}$ (because the empty set $\varnothing$ is the union of the empty family). So every topology has at least two bases.