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Results tagged with axiom-of-choice
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user 11640
An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.
23
votes
Accepted
Axiom of choice as zero dimensionality
Here are two possible motivating examples.
First, for any topos $\mathcal{E}$, if all epimorphisms in $\mathcal{E}$ split, then $\mathcal{E}$ is a boolean topos. In particular, for a topological spa …
- 15.9k
18
votes
Accepted
Category and the axiom of choice
Here's a somewhat trivial one, but it is one that category theorists use all the time:
Let us say that a functor $F : \mathcal{C} \to \mathcal{D}$ is a weak equivalence if it is fully faithful and …
- 15.9k