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Suppose you take mathematical structures which have axioms based on sets and their subsets and you replace this with objects and subobjects, for example:

Let a very general topological space T be an object A together with a collection T of subobjects of A satisfying:

1) The base object and A are in T

2) If G and H are in T then so is the greatest common subobject of G and H

3) For any subobjects Oi in T the lowest common superobject of the Oi is also in T

Where greatest common subobject and lowest common superobject exist and can be uniquely defined, and base object B is defined to be the unique object, where such exists, such that greatest common subobject(B,A)=B for all A and lowest common superobject(B,A)=A for all A, depending on some specified composition/decomposition for these objects.

Between topological spaces, continuity of maps at a point x could be rephrased as continuity on the subset {x} in order to generalise to continuity at a subobject G.

Has this sort of thing been done before ?

Edit:

Also what about very general sigma-algebras: the complement of subobj G in A would be the lcm of all the subobjs of A that don't have G as a subobj. Or very general matroids ? : M is an object E with a collection I of independent subobjects such that: 1) base object is in I, 2) If A is in I, then every subobj of A is in I, 3) For A and B in I, if A is in some way larger than B then there is a subobj C of A that is not a subobj of B such that lcm (B,C) is in I.

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I don't know of something exactly like that, but what you write does somewhat resemble pointless topology, which basically studies the lattice of open sets on its own merits. You can take a look at this survey by Johnstone: projecteuclid.org/… –  Michael Greinecker Nov 10 '11 at 11:45

3 Answers 3

up vote 13 down vote accepted

Yes, this is basically the idea behind pointfree topology. Abstractly, the lattice of open sets is a frame, that is a complete lattice that satisfies the distributive law $$b \wedge \bigvee_{i \in I} a_i = \bigvee_{i \in I} b \wedge a_i.$$ The category of locales (pointfree spaces) is the dual of the category of frames (because a continuous map $f:X\to Y$ maps open subsets $U$ of $Y$ to open subsets $f^{-1}(U)$ of $X$).

There is an obvious functor from the category of topological spaces to the category of locales which associates to every space its frame of open sets. However, this functor is not an equivalence of categories. Many spaces can have the same lattice of open sets (e.g. all indiscrete spaces correspond to the two-element lattice) and some frames do not correspond to the lattice of open sets of some topological space.

The standard introductory reference for pointfree topology is Peter T. Johnstone's Stone Spaces.

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+1 because you call it "pointfree topology" and don't use the "pointless" pun. –  Qfwfq Nov 10 '11 at 15:00
    
The word for topological spaces that do behave well with respect to the embedding in the category of locales is sober spaces if I remember correctly. –  Qfwfq Nov 10 '11 at 15:04
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If you prefer something 'in the nature of a trailer', Johnstone also wrote 'The Point of Pointless Topology' ams.org/journals/bull/1983-08-01/S0273-0979-1983-15080-2/…. –  David Corfield Nov 10 '11 at 15:09
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@Qfwfq: But "Pointless topology" is such an awesome name. I mean, what's the point of doing mathematics if you cannot do it hilariously? :-) –  Asaf Karagila Nov 10 '11 at 15:59
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@AK: good point. thanks for pointing it out... –  Qfwfq Nov 10 '11 at 19:46

I would have a look at Grothendieck topologies e.g. http://en.wikipedia.org/wiki/Grothendieck_topology

It might not be the very same thing you describe here. But as you ask in such a general way this might still be satisfactory for you.

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In topological groups (rings,modules) you only have to say which subgroups (ideals, submodules) are open to determine the topology, sort of examples.

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You may be interested in contributing to a proposed Spanish language version of math stackexchange; it could use some input from fluent professors and students: area51.stackexchange.com/proposals/64529/… –  Brian Rushton Feb 2 at 20:51

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