Axioms of separation in semitopological groups and related functors
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文摘
We prove that for every semitopological group G and every , there exists a continuous homomorphism onto a (resp., for ) semitopological group H such that for every continuous mapping to a - (resp., - for ) space X, one can find a continuous mapping satisfying . In other words, the semitopological group is a -reflection of G. It turns out that all -reflections of G are topologically isomorphic. These facts establish the existence of the covariant functors for , as well as the functors Reg and Tych in the category of semitopological groups and their continuous homomorphisms.
We also show that the canonical homomorphisms of G onto are open for and provide an internal description of the groups and by finding the exact form of the kernels of and . It is also established that the functors Reg and , for are naturally equivalent.