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AngryParsley comments on Open Thread: January 2010 - Less Wrong
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P(A)*P(B|A) = P(B)*P(A|B). Therefore, P(A|B) = P(A)*P(B|A) / P(B). Therefore, woe is you should you assign a probability of 0 to B, only for B to actually happen later on; P(A|B) would include a division by 0.
Once upon a time, there was a Bayesian named Rho. Rho had such good eyesight that she could see the exact location of a single point. Disaster struck, however, when Rho accidentally threw a dart, its shaft so thin that its intersection with a perfect dartboard would be a single point, at a perfect dartboard. You see, when you randomly select a point from a region, the probability of selecting each point is 0. Nonetheless, a point was selected, and Rho saw which point it was; an event of probability 0 occurred. As Peter de Blanc said, Rho instantly fell to the very bottom layer of Bayesian hell.
Or did she?
Don't worry, the mathematicians have already covered this.
There are mathematicians who have rejected the idea of the real number line being made of points, perhaps for reasons like this. I don't recall who, but pointless topology mght be relevant.
My understanding is that such a story relies on trying to define the area of a point when only areas of regions are well-defined; the probability of the point case is just the limit of the probability of the region case, in which case there is technically no zero probability involved.
Is pointless topology ever relevant?
Yes, it is relevant to algebraic geometry, which is important for the treatment of down-to-earth problems in number theory.
I think you're confusing topos theory with pointless topology. The latter is a fragment of the former and a different fragment is used in algebraic geometry. As I understand it, the main point of pointless topology is to rephrase arguments to avoid the use of the axiom of choice (which is needed to choose points). That is certainly a noble goal and relevant to down-to-earth problems, but not so many in number theory.