We develop the theory of Heyting locally small spaces, including Stone-like dualities such as a new version Esakia duality and system concrete isomorphisms equivalences. In way, we continue building tame topology, realising Grothendieck’s ideas. use up-spectral spaces define standard up-spectralification Kolmogorov space. This research gives more understanding definable over structures with top...

This paper presents an abstract parametrised functional interpretation of WE - HA ω . It is based on families parameters allowing for different degrees freedom the design concrete interpretations. In this way, we are able to generalise previous work unifying interpretations by including in unification more recent bounded and Herbrandized

For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone...

The appearance of the complete Heyting algebra in the realm of Algebraic Topology is the main topic of the paper.