Why does the definition of topological space require that the empty set be open?

Technically we could define topology without the empty set. But then everywhere an empty set appears we would have to deal with it in a special way. For example a continuous function $f:X\to Y$ is a function such that preimage of any open set is open. Without the "$\emptyset\in\tau$" assumption we would have to modify this definition: preimage of every open set is open if nonempty. And this special "if nonempty" now has to appear everywhere.

This is an unnecessary noise. Makes proofs and definitions longer, less readable and for no good reason.

Also note that similar reasoning applies to "$T\in\tau$". Since we often deal with closed sets which by definition are complements of open sets. We now have to deal with two special cases: "if nonempty" and "if not entire $T$". Which can be avoided by two simple axioms.

First of all, in your examples $\emptyset$ is not a set with a compact complement, as $\Bbb R^n$ is not compact, and it does not have a finite complement $[0,1]$ in the second case.. So they should be explicily added to $\tau$ as one of the axioms states it should be in there.

Also, in many cases, like topologies from metric spaces or ordered spaces, there will be many cases where the topology contains disjoint sets and we want open sets to be closed under finite intersections, so for that reason alone we have to add $\emptyset$ to $\tau$ or accept that a topology reduces to a weakened sort of filter on $X$ (another type of families of subsets, also interesting, but not topology; your second example as stated (without the empty set) is indeed a filter BTW and the first a so-called filter-base ). Also formally, in the arbitrary union axiom we could take $I=\emptyset$ and then we're also "forced" to add $\emptyset$ to $\tau$ in order to fulfill it.

In short, we have the empty set in $\tau$ for practical and internal logic/consistency reasons. We consider mostly non-empty open sets in practice (as these are the neighbourhoods used in continuity/convergence etc.), but we do need it in there.

This question has many more reasons in its answers.

Taking a categorical perspective we should ask why are we even defining topological spaces. In a category we are primarily interested in the morphisms and the objects are merely there to serve as domains and codomains. So, the reason we define topological spaces is in order to define continuous functions between them. A function $f\colon X\to Y$ is continuous if it pulls back every open set in $Y$ to an open set in $Y$. Namely once topologies have been chosen, for $f$ to be continuous it must be the case that $f^{-1}(U)$, the inverse image of $U\subseteq Y$, is open in $X$. At first site this may seem like an odd definition but it just justified in many ways, chiefly, at the elementary level, by the desire to mimic the usual $\varepsilon - \delta $ notion of continuity which also goes in reverse. In any case, one certainly wants all constant functions to be continuous. If we did not demand the empty set to always be open, then constant functions would typically fail to be continuous since you can find, typically, plenty of open sets in the codomain whose inver image is empty.