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The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with rigor).This got me thinking, could this be done with topology to create hypertopology (an equivalent but perhaps more intuitive way of doing topology)?

It should be able to translate, since hyperreals can be represented as ultrafilters of reals. Hypertopology would involve ultrafilters of the points of the topology presumably.

To define the topology, I presume you would define when two points of the hypertopology are infinitesimally close. To convert a metric space into a topology, you would simply define two hypertopology points as close whenever their distance is infinitesimal (again, ultrafilters can be applied point wise.)

Has this ever been studied? What axioms would hypertopology "closeness" need to follow to be equivalent to regular topology? Is there an axiomatic approach (not requiring the ultrafilters (the hyperreals have an axiomatic basis))?

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    $\begingroup$ Chapter 2 of Albeverio's Nonstandard methods in stochastic analysis and mathematical physics in on topology and linear spaces in the context of nonstandard analysis. $\endgroup$
    – Chappers
    Oct 3 '15 at 1:12
  • $\begingroup$ Since any hyperreal field is an orderded field, it can be equipped with the order topology. $\endgroup$
    – nombre
    Oct 3 '15 at 11:05
  • $\begingroup$ @nombre I'm not talking about topologies on hyperstuff. I'm talking about using the hyperversion of something to define its topology. $\endgroup$
    – PyRulez
    Oct 3 '15 at 11:11
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    $\begingroup$ @nombre I want to generalize this beyond the reals to as much topology as possible. As hyperreals replaced limits with infinitesimals, hypertopology should replace neighborhood arguments with infinitely close point arguments. $\endgroup$
    – PyRulez
    Oct 3 '15 at 12:08
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    $\begingroup$ Non-standard topology normally starts with a topological space $\langle X,\tau\rangle$ and forms a non-standard extension ${^*X}$, so you have the topology on $X$ to begin with. However, you can then use the extension to prove things about $X$ just as you can use the hyperreals to prove things about $\Bbb R$. This PDF of an M.Sc. thesis at the Islamic Univ. of Gaza gives an introduction to the notions. $\endgroup$ Oct 3 '15 at 17:31
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Non-standard topology normally starts with a topological space $\langle X,\tau\rangle$ and forms a non-standard extension ${^*X}$, so you have the topology on $X$ to begin with. However, you can then use the extension to prove things about $X$ just as you can use the hyperreals to prove things about $\Bbb R$. This PDF of an M.Sc. thesis at the Islamic University of Gaza gives an introduction to the notions.

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As PyRulez suggested, the hyperreals are useful in clarifying certain topological concepts. For example, compactness of $X$ can be characterized by saying that every point of ${}^\ast\! X$ is infinitely close to a standard point of $X$. Using this characterisation dramatically shortens proofs of key results, such as compactness of a product of compact spaces.

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