When do two subbases generate the same topology

Question

Let $X$ be a set.

If $\mathcal B_1$ and $\mathcal B_2$ are bases of subsets of $X$, it is well-known that $\mathcal B_1$ and $\mathcal B_2$ generate the same topology if and only if for any pair of basic sets $B_1, B_2 \in \mathcal B_i$ and $x \in B_1 \cap B_2$ there exists $B_3 \in \mathcal B_j$ with $ x \in B_3 \subset B_1 \cap B_2$, where $\{i,j\} = \{1,2\}$.

Is there a similar criterion to check whether two subbases generate the same topology?

Recall:

A base is a family $\mathcal B$ of subsets of $X$ with the property that every $x \in X$ belongs to at least one $B \in \mathcal B$, and for any $B_1,B_2 \in \mathcal B$ and $x \in B_1 \cap B_2$ there exists $B_3 \in \mathcal B$ with $x \in B_3 \subset B_1 \cap B_2$.

A subbase is a family $\mathcal S$ of subsets of $X$ whose members form a cover of $X$. The family of finite intersections of members of $\mathcal S$ forms a basis.

It would be that the bases generated by each subbase generate the same topology on $X$ — R_D, Feb 12 '16 at 15:16
Of course it would. I'm wondering if there is a simple criterion that just compares elements of the subbases. — user312938, Feb 12 '16 at 15:19
Your actual criterion for base is a little too convoluted. Check this answer out. — Pedro Sánchez Terraf, Feb 12 '16 at 20:12

score 2 · Accepted Answer · answered Feb 12 '16 at 20:12

$\newcommand{\S}{\mathcal{S}}$ One criterion is the following: Two subbases $\S$ and $\S'$ generate the same topology iff for every $A\in\S$ and $x\in A$ there are $A_1',\dots, A_n'\in\S'$ such that $x\in A_1'\cap\dots \cap A_n'\subseteq A$, and conversely. I think you won't find something simpler.

When do two subbases generate the same topology

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