Definition of the subbasis : Don't we have to include the empty set?

Question

Let $X$ a set and $\mathcal A=\{A_1,...,A_n\}\subset 2^X$. I want to construct the thickest topology such that elements of $\mathcal A$ are open. In my course it's written that a basis for $\mathcal T$ is $$\left\{\bigcap_{i\in I}A_i\mid |I|<\infty \right\}\cup \{X\},$$ or we can also set $$\mathcal B=\mathcal A\cup\{X\},$$ and a basis is given by $$\left\{\bigcap_{i\in I}A_i\mid |I|<\infty \right\}.$$

My question : if the $A_i$'s are not disjoint, don't we have to add the empty set ? i.e. we set $\mathcal B$ as $$\mathcal B=\mathcal A\cup\{\emptyset, X\},$$ instead of $\mathcal A\cup \{X\}$ ? I asked an tutor by email, and he just answered me that it was not necessary. Why it's not ? How can we get it ? I really don't understand.

Answer 1

You don't explain why you think you need to include the empty set, so it's hard to answer.

However, I imagine your issue is that you haven't internalized the following, or something like it.

When we say "the open sets are the unions of basis elements", that means unions of all possible shapes, such as:

• A union of an infinite family of basis elements
• The union of a large but finite family of basis elements
• The union of a pair of elements; i.e. the union of a set containing two basis elements: e.g. $U \cup V = \bigcup_{S \in \{U, V\}} S$
• The union of a family consisting of just one basis element: e.g. $U = \bigcup_{S \in \{U\}} S$
• The union of an empty family of basis elements: e.g. $\varnothing = \bigcup_{S \in \varnothing} S$

In particular, the empty set can be written as a union of basis elements.

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It's maybe worth noting that your textbook has made a similar oversight; it's redundant to explicitly include $\{ X \}$ in the basis, because their definition already includes it: the $I=\varnothing$ case is the empty intersection

$$ X = \bigcap_{i \in \varnothing} A_i $$

_{(technical note: this formula for an empty intersection implicitly depends on the context that we are doing arithmetic with "subsets of $X$")}