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Even Continuity And Topological Equicontinuity In Topologized Semigroups

机译：拓扑半群中的偶连续性和拓扑等连续性

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摘要

A topologized semigroup X having an evenly continuous resp., topologically equicontinu-ous, family R_x of right translations is investigated. It is shown that: (1) every left semi-topological semigroup X with an evenly continuous family R_x is a topological semigroup, (2) a semitopological group X is a paratopological group if and only if the family R_x is evenly continuous and (3) a semitopological group X is a topological group if and only if the family R_x is topologically equicontinuous. In particular, we get that for any paratopological group X which is not a topological group, the family R_x provides an example of a transitive group of homeomorphisms of X that is evenly continuous and not topologically equicontinuous. The last conclusion answers negatively a question posed by H.L. Royden.
机译：研究了拓扑正确的半群X，该半群X具有正确连续的，拓扑上连续的右翻译族R_x。结果表明：（1）每个左半拓扑X具有均等连续的族R_x是拓扑半群；（2）半拓扑群X当且仅当R_x族均匀连续且是（3） ）当且仅当族R_x在拓扑上是等连续的时，半拓扑组X才是拓扑组。特别是，我们得到了对于不是拓扑组的任何拓扑X族，R_x族提供了X个同态同构的传递组的一个例子，该传递同构是均匀连续的，而不是拓扑上等连续的。最后的结论否定了罗伊登（H.L. Royden）提出的问题。

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