Topology 101 is not about holes. It is about open sets and continuous functions.
That said, any mathematician calling themselves a “topologist” is likely studying things like homology/homotopy or generalizations which mostly apply to Euclideanish spaces.
Good point! I guess it's not different to the equator on the surface of a sphere... so yes, a straight line "encloses" half of the plane "inside" it, so it has a hole.
Of course, my definition only works for the type of wholes that are cavities. Defining other types of holes, as discussed in the article, requires more than elementary point-set topology.
My topology is rusty, but I do not think that a line in the plane is equivalent to an equator of a sphere because a plane is not equivalent to a sphere in a topological sense, and neither is a line equivalent to a circle. There is no homeomorphism (structure preserving bijective function) between either pairs of spaces.
There is a standard way to augment the plane/line to make it equivalent to a sphere/circle called the "1-point compactification"[1], but since it requires adding an extra point "at infinity", the augmented space is not the same as the original.
So no, a straight line doesn't have a "hole" in the topological sense. The 1-point compactification of it does though.
Sure the definition of a hole still works (as does the more proper definition) but whether it is remotely usable is a different matter.
What makes it dead?
That is to say it's an essential part of the foundations of theoretical mathematics (you can't understand algebraic topology without understanding point-set topology), but there is not much new innovation purely within point-set topology itself.
So I find "dead as a dodo" too strong of a metaphor.
The most common meaning of the word "topology" is quite simply "a topology". As in a set of opens for a particular space of interest. These show up all over the place regardless of whether you can do some algebra on them.