On the nonsimplicity of some convergence categories

Abstract

Abstract.The category TOP of topological spaces is known to be Convsimple in the sense that there exists a single object E in Top such that Top is the epireflective hull of {£} in the category Conv of convergence spaces. Prtop, the category of pretopological spaces is also Conv-simple. We show that on the contrary the category Pstop of pseudo topological spaces and Conv itself are not Conv-simple. More specifically every epireflective subcategory of Conv which contains all Hausdorff oembedded locally compact spaces is not Conv-simple.For references on topological categories, reflective subcategories and reflective hulls we refer to [6,8,9,10].Let sé be a topological category. All subcategories 38 are assumed to be full and isomorphim-closed. A subcategory 38 of sé is epireflective in sé if it is closed with respect to the formation of products and subobjects in sé . In this context " Y is a subobject of X " means that there exists an embedding from X to Y . This notion coincides with the categorical notion of extremal subobject.Every subcategory < §* of sé is contained in a smallest epireflective subcategory, its epireflective hull, which is denoted by R%. An object A of sé belongs to R% if and only if A is a subobject of a product of objects of <£ . A subcategory 33 of sé is called sé -simple if there exists a single object E of 38 such that 38 is the epireflective hull in sé of the class {E} , i.e., 38 = R{E} .Several examples of this situation are well known. If we take sé = Conv then its bireflective subcategory Top is Conv-simple. Several subcategories of TOP are Conv-simple too, see for instance [4,5,6,7,9,14]. Simplicity remains true if TOP is enlarged to the bireflective subcategory Prtop. This can be derived from results in [1].In this paper we show that simplicity however does not extend to the larger bireflective subcategories Pstop or to Conv itself.

Three problems in convergence theory

“…The same is true for pretopologies ξ when we consider maps from ξ to the Bourdaud pretopology (see Antoine [1] and Bourdaud [2,3]). However, as proved by Eva and Robert Lowen [16], there is no initially dense pseudotopology and the same negative result holds for the class of all convergences. In this paper we will investigate this property for the classes of paratopologies and hypotopologies.…”

Simplicity of Categories Defined by Symmetry Axioms

“…Whenever F is a convergence space let -0 Y be its Pretop reflection. The neighborhoodfilters of 0 Y are given by The result of Theorem 2.3 should be compared with the next one which was obtained by R. Lowen together with the first author in [14]. THEOREM 2.5 [14].…”

“…PROOF. The proof is analogous to the proof of the main theorem in [14]. Let Y be an arbitrary filter space.…”

“…The neighborhoodfilters of 0 Y are given by The result of Theorem 2.3 should be compared with the next one which was obtained by R. Lowen together with the first author in [14]. THEOREM 2.5 [14]. For every convergence space Y there exists a T\ c-embedded convergence space X such that for any source (/ : X -> Y) ie j the space X is not initial in Conv.…”

“…Now let (f: X -• » Y) ieI be a source in Fil, then by the proposition in [14] for every / G / there exists an ultrafilter Wj ^ 11 such that stacky fCWi) -stacky fi(ll). Hence stocky fi (11) is micromeric for every / G /.…”