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Nets (and also filters) allow one to unify the wide morass of convergence notions. There is even an undergraduate introduction to analysis based on nets, Limits by Alan Beardon.

Notions like the convergence of sets (the liminf and limsup of sets stuff) can be formulated as a nontopological form of order convergence based on nets. These notions play an important part in general measure theory and especially in probability theory.

Kolmogorov's extension theorem of stochastic processes has abstract generalizations based on nets of $\sigma$-algebras. See for example here (Wayback Machine).

Ultrafilters play a surprising role in social choice theory. The family of "pivotal sets of voters" implied by a social decision rule satisfying the assumption of Arrow's impossibility theorem forms an ultrafilter, which explains why all such rules are dictorial with finitely many voters but not in the case of infinitely many voters.

For an actual application of ultrafilters in geometry, see this post on Terry Tao's blog.

Nets (and also filters) allow one to unify the wide morass of convergence notions. There is even an undergraduate introduction to analysis based on nets, Limits by Alan Beardon.

Notions like the convergence of sets (the liminf and limsup of sets stuff) can be formulated as a nontopological form of order convergence based on nets. These notions play an important part in general measure theory and especially in probability theory.

Kolmogorov's extension theorem of stochastic processes has abstract generalizations based on nets of $\sigma$-algebras. See for example here (Wayback Machine).

Ultrafilters play a surprising role in social choice theory. The family of "pivotal sets of voters" implied by a social decision rule satisfying the assumption of Arrow's impossibility theorem forms an ultrafilter, which explains why all such rules are dictatorial with finitely many voters but not in the case of infinitely many voters.

For an actual application of ultrafilters in geometry, see this post on Terry Tao's blog.

Nets (and also filters) allow one to unify the wide morass of convergence notions. There is even an undergraduate introduction to analysis based on nets, Limits by Alan Beardon.

Notions like the convergence of sets (the liminf and limsup of sets stuff) can be formulated as a nontopological form of order convergence based on nets. These notions play an important part in general measure theory and especially in probability theory.

Kolmogorov's extension theorem of stochastic processes has abstract generalizations based on nets of $\sigma$-algebras. See for example here.

Ultrafilters play a surprising role in social choice theory. The family of "pivotal sets of voters" implied by a social decision rule satisfying the assumption of Arrow's impossibility theorem forms an ultrafilter, which explains why all such rules are dictorial with finitely many voters but not in the case of infinitely many voters.

For an actual application of ultrafilters in geometry, see this post on Terry Tao's blog.

Nets (and also filters) allow one to unify the wide morass of convergence notions. There is even an undergraduate introduction to analysis based on nets, Limits by Alan Beardon.

Notions like the convergence of sets (the liminf and limsup of sets stuff) can be formulated as a nontopological form of order convergence based on nets. These notions play an important part in general measure theory and especially in probability theory.

Kolmogorov's extension theorem of stochastic processes has abstract generalizations based on nets of $\sigma$-algebras. See for example here (Wayback Machine).

Ultrafilters play a surprising role in social choice theory. The family of "pivotal sets of voters" implied by a social decision rule satisfying the assumption of Arrow's impossibility theorem forms an ultrafilter, which explains why all such rules are dictorial with finitely many voters but not in the case of infinitely many voters.

For an actual application of ultrafilters in geometry, see this post on Terry Tao's blog.

Nets (and also filters) allow one to unify the wide morass of convergence notions. There is even an undergaraduate introduction to analysis based on nets, Limits by Alan Beardon.

Notions like the convergence of sets (the liminf and limsup of sets stuff) can be formulated as a nontopological form of order convergence based on nets. These notions play an important part in general measure theory and especially in probability theory.

Kolmogorov's extension theorem of stochastic processes has abstract generalizations based on nets of $\sigma$-algebras. See for example here.

Ultrafilters play a surprising role in social choice theory. The family of "pivotal sets of voters" implied by a social decision rule satisfying the assumption of Arrow's impossibility theorem forms an ultrafilter, which explains why all such rules are dictorial with finitely many voters but not in the case of infinitely many voters.

For an actual application of ultrafilters in geometry, see this post on Terry Tao's blog.

Nets (and also filters) allow one to unify the wide morass of convergence notions. There is even an undergraduate introduction to analysis based on nets, Limits by Alan Beardon.

Notions like the convergence of sets (the liminf and limsup of sets stuff) can be formulated as a nontopological form of order convergence based on nets. These notions play an important part in general measure theory and especially in probability theory.

Kolmogorov's extension theorem of stochastic processes has abstract generalizations based on nets of $\sigma$-algebras. See for example here.

Ultrafilters play a surprising role in social choice theory. The family of "pivotal sets of voters" implied by a social decision rule satisfying the assumption of Arrow's impossibility theorem forms an ultrafilter, which explains why all such rules are dictorial with finitely many voters but not in the case of infinitely many voters.

For an actual application of ultrafilters in geometry, see this post on Terry Tao's blog.