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The fundamental category of Topological spaces (without the Hausdorff property) is the "Sobre" Sober" spaces category, this is strictly related to Zarisky Zariski topology, and to locales and Topos (growing in generalization). See EGA1 (Grothendieck, Dieudonne, Springer), or "Stone SPaceSpaces" of P. Johnstone.

Of course Hausdorff Topological spaces are related (roughly) to a our usual way of see the geometrical spaces, in a non Hausdorff space points are related for other intrinsic (logical, geometrical, algebraic, orders) criteria, then is right that our usually intuitive point of view lack to rapresentate represent them. But when we escape from Hausdorff propriety we are near to escape from "space as set of points" concept, see the concept of locales or frames ("Stone spacesSpaces" Johnstone).
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The foundamental fundamental category of Topological spaces (out without the Hausdorff proprietyproperty) is the "Sobre" spaces category , this strict is strictly related to Zarisky topology, and to locales and Topos (growing in generalization). See EGA1 (Grothendieck, Dieudonne, Springer), or "Stone SPace" of P. Johnstone.

Of course Hausdorff Topological space spaces are related (roughtly) roughly) to a our usual way of see the geometrical spaces, in a non Hauasdorff spaces Hausdorff space points are related for other intrinsic (logical, geometriclgeometrical, algebraic, orders) criteria, then is right that our usually intuitive point of view lack to rapresentate them. But when we esacape escape from Hausdorff propriety we are near to escape from "space as set of points" concept, see the concept of locales or frames ("Stone spaces" Johnstone).
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The foundamental category of Topological spaces (out the Hausdorff propriety) is the "Sobre" spaces category , this strict related to Zarisky topology, and to locales and Topos (growing in generalization). See EGA1 (Grothendieck, Dieudonne, Springer), or "Stone SPace" of P. Johnstone.

Of course Hausdorff Topological space are related (roughtly) to a our usual way of see the geometrical spaces, in a non Hauasdorff spaces points are related for other intrinsic (logical, geometricl, algebraic, orders) criteria, then is right that our usually intuitive point of view lack to rapresentate them. But when we esacape from Hausdorff propriety we are near to escape from "space as set of points" concept, see the concept of locales or frames ("Stone spaces" Johnstone).