For more on loclaes
For all the locale stuff, Vickers is pedagogical
https://www.cs.bham.ac.uk/~sjv/the first main article is
LOCALES AND TOPOSES AS SPACES
https://www.cs.bham.ac.uk/~sjv/LocTopSpaces.pdfhttps://warosu.org/sci/thread/S12166845#p12170887so take the singleton {*}
and the poset of 2 elements 2
the frame of {*} this is ?(1)
you have the equivalence of sets ?(1) ? Idl(2) = ideals on poset 2
the poset 2 has a topology, ie the frame
?(2) ? Idl(4) = ideals on poset 4
now the sierpinski locale:
take the poset 2
take the ideal completion of the poset 2
put the scott topology on Idl(2)
this is the frame of the ''sierpinski space'' S
theorem:
?(S) ? Alex(2)= alexendrov topology on 2 ? Filters on 3
now this sierpinski space is a classifying topos, ie how to get points of spaces. which is why it matters. When you manipulates points of a space in a topos, you work with the Sierpinski (or some generalization of it)
>The Sierpinski topos is the classifying topos for subterminal objects in toposes (see e.g. Johnstone 77, p. 117).https://ncatlab.org/nlab/show/Sierpinski+toposMore on this space, as a ASD theory
https://www.paultaylor.eu/~pt/ASD/foufct/sierpinski.html 
