Interplay between algebraic and topological closure operators, and existence of compact-like group topologies on abelian groups

刻划了尺寸至多为连续统幂集的交换群G的蕴稠子集，并证明了这样一个群G的子集S的马尔可夫-扎里斯基闭包可以在G上的某种预紧群拓扑中实现(特别地，G中的集合的潜在稠密性可以用G的某种预紧群拓扑来证明)。得到了COMPCAT群的新的度量化准则。通过控制李群的零维闭子群，给出了李群的各种刻画。发展了拓扑群的乘子收敛理论。

We characterize potentially dense subsets of abelian groups G of size at most the power set of the continuum, and we show that the Markov-Zariski closure of a subset S of such a group G can be realized in some precompact group topology on G. (In particular, potential denseness of a set in G can be witnessed by some precompact group topology of G.) New metrizability criteria for compcat groups are obtained. We also provide various characterizations of Lie groups by means of controlling their zero-dimensional closed subgroups. Multiplier convergence theory of topological groups is developed.