Extensive Subcategories of the Category of T1-spaces

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Abstract
It is well known that epimorphisms in the category Top (Top1, respectively) of topological spaces (T1spaces, respectively) and continuous maps are precisely onto continuous maps. Since every mono-reflective subcategory of a category is also epi-reflective and every embedding in Top (Top1, respectively) is a monomorphism, there is no nontrivial reflective subcategory of Top (Top1 respectively) such that every reflection is an embedding. However, in the category Top0 of T0-spaces and continuous maps as well as in the category Haus of Hausdorff spaces and continuous maps, there are epimorphisms which are not onto. Moreover, every reflection of a reflective subcategory of Top0, which contains a non T1-space, is an embedding [16].

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