Subset-system derived topologies and coreflective categories of T0-spaces
Abstract
Given a subset system H on T0-spaces, we associate with each topology a closure-testing derived topology and study the fixed points of this operator. The H-determined topologies above a given topology are closed under intersections and hence determine a least fixed-point hull. We prove that this hull is obtained by transfinite iteration, is functorial, and defines a right adjoint to the inclusion of H-determined spaces into Top0. Consequently, the corresponding full subcategory is coreflective and complete. We isolate restriction-stability and open-locality hypotheses under which determinedness is inherited by closed and open subspaces, and we establish a general non-reflectivity criterion. If H is contained in the directed-subset system, the category of H-determined spaces is Cartesian closed; its exponentials are obtained by coreflecting the pointwise convergence topology on continuous maps. The construction subsumes familiar monotone- and irreducible-determined cases while separating the formal consequences of the subset-system axioms from the additional hypotheses needed for heredity and exponentials. It therefore provides a uniform categorical mechanism for producing complete Cartesian closed categories of T0-spaces with explicit higher-order function spaces, as required in denotational semantics.
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BibTeX
@misc{xu2026subsetsystem,
title = {{Subset-system derived topologies and coreflective categories of $T_0$-spaces}},
author = {Xiaoquan Xu and Xinjie Huang and Yi Yang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01698},
eprint = {2611.01698},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01698},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Subset-system derived topologies and coreflective categories of T_0-spaces
AU - Xiaoquan Xu
AU - Xinjie Huang
AU - Yi Yang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01698
UR - https://arxiv.org/abs/2611.01698
AB - Given a subset system H on T_0-spaces, we associate with each topology a closure-testing derived topology and study the fixed points of this operator. The H-determined topologies above a given topology are closed under intersections and hence determine a least fixed-point hull. We prove that this hull is obtained by transfinite iteration, is functorial, and defines a right adjoint to the inclusion of H-determined spaces into Top_0. Consequently, the corresponding full subcategory is coreflective and complete. We isolate restriction-stability and open-locality hypotheses under which determinedness is inherited by closed and open subspaces, and we establish a general non-reflectivity criterion. If H is contained in the directed-subset system, the category of H-determined spaces is Cartesian closed; its exponentials are obtained by coreflecting the pointwise convergence topology on continuous maps. The construction subsumes familiar monotone- and irreducible-determined cases while separating the formal consequences of the subset-system axioms from the additional hypotheses needed for heredity and exponentials. It therefore provides a uniform categorical mechanism for producing complete Cartesian closed categories of T_0-spaces with explicit higher-order function spaces, as required in denotational semantics.
ER -