my book says that "... If we wish to emphasize the topology T, then we refer to U as T-open, ..."
Does this then mean that U fullfils the topology properties?
It means that U is an open set in the topology T. It's the topology T that has to fulfil the topology properties. The set U just has to satisfy the condition of being one of the open sets in that topology.
It means that U is an open set in the topology T. It's the topology T that has to fulfil the topology properties. The set U just has to satisfy the condition of being one of the open sets in that topology.