Topologies and sigma-algebras as "hypergraphs" containing an "edge" having 0 endpoints A hypergraph $H$ is a pair $H=(X,E)$ where $X$ is a set of elements called nodes and $E$ is a set of non-empty subsets of $X$ called hyperedges.
I'm wondering about the motivation behind specifying that hyperedges cannot have 0 endpoints ($\emptyset$ is never a hyperedge)?
If $\emptyset$ is allowed to be a hyperedge, then things like topologies and $\sigma$-algebras on sets become examples of hypergraphs, which makes a relaxed definition of hypergraph seem like a nice abstraction. Does allowing $\emptyset \in E$ have significant consequences?
 A: There is no reason to assume that $\emptyset$ cannot be an edge. If you found a source that does, it is probably taking a cue from Berge's Graphs and Hypergraphs, which is one of the earlier textbooks that discusses hypergraphs in detail.
Berge forbids the empty set as an edge, but also forbids isolated vertices: the union of all edges must be the vertex set. If we add one of these conditions, it makes sense to add the other, because they are dual conditions (if we reverse the roles of vertices and edges) and Berge immediately goes on to discuss dual hypergraphs.
On the other hand, Bollobás's Modern Graph Theory does not require all edges of a hypergraph to be nonempty, and so if you'd like to follow suit, you're in good company.
A: It's mostly a convention given in some books but it's not mandatory. There are some conventions on graphs which differ in the literature, which makes it troublesome sometimes, so other areas of mathematics tend to use their own terms. For example, "quiver" in representation theory means what I'd call a "multidigraph" which in some of the literature is just a "digraph". A simple digraph would be a digraph without repeated edges or loops, but for me a simple digraph is what I call a digraph. Everyone in representation theory calls it a quiver, though.
So, I think these are conventions to use, and change carefully when you need to. I work on some properties of hypergraphs and certain simplicial complexes related to them, and I like to see a simplicial complex as an hypergraph with some other properties, one of which is that $\emptyset$ is an edge.
In hypergraphs different from simplicial complexes I'm not really interested when $\emptyset$ is an edge, since the associated simplicial complex will end up being a simplex, though. I'm also not interested in isolated vertices since they also correspond to a complex without homology, which also is how Berge defines them. I don't go too far, though. I define them very generally, but only work on those cases.
For the case of $\sigma$-algebras and topologies you mention, well, it's usually prefered when vertex set is finite and to be finitely many edges which are also finite, but of course, there are always exceptions. I don't know about cases when everything is allowed to be infinite, but when we work on an "infinite" simplicial complex we prefer faces (edges of the corresponding hypergraph) to be finite, even if there are infinitely many faces and vertices. The theory fits better if everything is finite (and can completely break if we allow faces with infinitely many elements).
So... Does allowing $\emptyset\in E$  have significant consequences?
Yes, and no. Yes, because the theory of hypergraphs can be used to know about other similar structures, like matroids, simplicial complexes, etc.
No, because even if we don't allow $\emptyset$ to be an edge, we can still associate to each of these structures a hypergraph by removing the empty set from it. In the case of simplicial complexes, for example, all the information is stored in the set of the facets (maximal faces), which, by Berge's definition, would be an hypergraph.
