Nature of the elements of spacetime? I am learning about relativity and am not quite sure how to think of spacetime. From a mathematical perspective, spacetime is a manifold i.e. a topological space for which about any point there exists an open neighbourhood that is homeomorphic to an open subset of R^4.
However a topological space (and thus manifold) is fundamnetally, in a mathematician's definition, a set of elements, X with some additional structure. Namely, that certain elements of the power set of X belong to a set- the topology on X- which we say are 'the open subsets of X'.
Pure math out of the way, I am not sure how to think of the points in spacetime. I have heard many physicists refer to the elements of the speactime manifold as 'events'. Yet this does not seem correct to me. Generally- and in most usage other than when you explicitly ask a physicist what spacetime is made of- and 'event' is an 'occurance', like "spaceship emitting a light pulse", "spaceship absorbing a light pulse" etc. It seems that events are ascribed to a specific spacetime point. More rigorously, an 'event' is uniquely assigned to an element of the manifold.
The whole point of the topological view is that the spacetime manifold has an existence independent of anything occuring in it. But I am unsure whether my understanding of 'events' and 'spacetime elements' is correct. And if so, I still find it somewhat unsatisfactory and would like to know what a phyiscist regards as the nature of the spacetime points in which specific events don't occur. These points have an existence as spacetime ppoints to which an event could have been ascribed (I laugh at my use of the past tense!) but wasn't. What is the ontology of these elements from a physicist's perspective?
 A: An "event" in relativity is analogous to a "point in space". 
Let's restrict to a 1+1 Minkowski spacetime and a 2-dimensional Euclidean plane.
To locate an event, one can specify a time-coordinate and space coordinate (t,x),
just like (x,y) for a point on the plane.
Note that this "event" applies just as well to an ordinary position-vs-time graph in PHY 101 (which is also a non-euclidean space, defining the metric by using a wristwatch to define lengths of worldlines).
From this point of view, I'm not saying anything about what happens (or doesn't happen) at these various (t,x)-pairs. Just like in a Euclidean space where a certain point can be used to describe the intersection of two specific lines, a physicist might say "that event is where car A traveling constant velocity meets another car B traveling with a different constant velocity"... or "the event when car C emits a light ray". In Euclidean space, you could have points in the plane that have no particular special significance... so similarly for Minkowski spacetime.
From the spacetime viewpoint, there is no issue of "excluding events" in the sense that something isn't allowed to happen. Maybe with additional structures and other laws of physics, such things can be done... but not at the spacetime level.
Robert Geroch (in his General Relativity from A to B  ISBN 978-0226288642) starts off his first chapter with "By an event we mean an idealized occurrence in the physical world having extension in neither space nor time"... later suggesting a revision to "idealized potential occurrence". The rest of the chapter develops the spacetime viewpoint, occasionally discussing aspects of philosophy of science that might arise. Although the target audience was non-science majors, many of the viewpoints persist into Geroch's more advanced publications in relativity. I suggest that this might be a good starting point.
A: This was meant to be a comment but became too long...
Events are exactly what each point of space time is from a classical perspective.  An event occurs at a point in space and at some time.  That is not to say that something is happening everywhere all the time.  But the value in the description is that (1) we use this space-time structure to label what we experience as events and (2) that we use observed events to build the space-time structure in our neighborhood.  What they may represent in a QFT or even an QG sense is another story, but we still refer to them as events in the same way that we try to think of an electron in classical terms when we know better from QM.
From a physicist's perspective I think that defining our local space-time neighborhood using life events is more meaningful than just saying that Minkowski space is a good tool for labeling events.  Observation is what drives us in science.  Even when the theory seems mathematically obtuse to many scientists.
What is happening is that we as a community of scientists are trying to attribute our experience (the foundation of empiricism) to the mathematical construct we use to describe the relationships among our experiences.  That will change as our experience evolves, and theories are abandoned and replaced.  
We can think of an electron "experiencing" another particle and call that meeting an event.  Whether or not "consciousness" is required to define event is a different story.  But once this leap is made the designation of event holds even when describing the interference of two quantum fields on classical curved background space-time.
A: Your description of spacetime events seems to be perfectly in accordance with physicists, as shown by the comments and the other answers. However, searching in physics stack exchange, you will not find much insight about this problem, and as an example, my own reflections in this question were not much appreciated (I received 4 downvotes.)
You are certainly right that there are open questions on this issue : It is a fact that the manifold character of spacetime 1. is a mere assumption, and that 2. it is not harmonizable with quantum mechanics (the problem of quantum gravity).
The assumption is originating from the famous lecture of Minkowski in 1908 : 

"In order to leave nowhere a gaping void, we imagine to ourselves that
  something perceptible is existent at all places and at every moment.
  In order to avoid using the words matter or electricity, I will use
  the word substance for this 'some thing'."

Since then, nobody doubted in the correctness of the assumption.
"Events" which are no "particle events" are points of vacuum. However, SR defines only worldlines but not the vacuum between the worldlines. As vacuum points do not belong to a worldline, they have no time evolution, they are timeless points in R3 space. For example, you cannot presume that vacuum points in a Minkowski diagram are going upwards because this would suppose a preferred observer.
So, in summary, your description is in accordance with physicists, but your doubts seem from my personal point of view justified. But perhaps and by surprise somebody can provide you the missing answer.
A: 
I have heard many physicists refer to the elements of the speactime manifold as 'events'.

Yes, this is correct. As you said above a spacetime manifold is a set with some additional structure. The elements of the set are the events. 
This is exactly analogous with a manifold in standard Riemannian geometry where the elements of the Reimannian manifold are points. Consider, for example, the 2D manifold of the surface of the earth. Each element of the manifold is a point on the surface of the earth, just as each element of spacetime is an event. 

Generally- and in most usage other than when you explicitly ask a physicist what spacetime is made of- and 'event' is an 'occurance', like "spaceship emitting a light pulse", "spaceship absorbing a light pulse" etc.

This is more about identifying specific events rather than defining what events are. Consider again the manifold of the surface of the earth. There are basically two ways to identify a point in that manifold. One is by giving its latitude and longitude, and the other is by identifying some landmark or directions e.g. “the peak of Everest” or “60 miles due east of Moscow”. 
Statements like the one you describe are ways to identify an event by some “landmark”. 

would like to know what a phyiscist regards as the nature of the spacetime points in which specific events don't occur.

The same nature as other events in the manifold. The nature of the location 60 miles east of Moscow is the same as the nature of the location of Moscow in the earth manifold. It is probably less interesting to visit there, but the location exists just the same. Similarly for events where specific physical landmarks do not occur. They are less interesting, but not less of an event in the theory. 
