The quasitopos hull of the construct of closure spaces

Abstract

<p>In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.</p>

Exponentiality for the construct of affine sets

“…In a similar way as for CL in [6], one can prove that F is an isomorphism. From 4.5 it then follows that K * is the CCT-hull of SSET.…”

“…For a power-closed collection C of SSET-objects in X, we can prove analogously to CL [6] that (X, A, A) where A = {U ⊆ X | (U, B) ∈ C for someB} and A the final structure determined by the sink of inclusion maps (i : (U, B) → X) (U,B)∈C is a K * -object such that C = C X .…”

“…We know from [7] that quotients in SSET are productive. In fact we have the same situation as for the category CL [6]. For CL a cartesian closed extension was constructed in [6] using the method presented by J. Adámek and J. Reiterman in [2].…”

“…In fact we have the same situation as for the category CL [6]. For CL a cartesian closed extension was constructed in [6] using the method presented by J. Adámek and J. Reiterman in [2]. We first look if this method is also applicable to SSET.…”

“…For the construct TOP of topological spaces this problem is extensively studied in the literature, see for example [17,8,4,5,19]. For the subconstruct CL of closure spaces, the author studied cartesian closedness in [6]. To remedy these facts, topologists have applied various methods.…”