Recent content by linulysses

  1. L

    Graduate Given Any Measurable Space, Is There Always a Topological Space Generating it?

    Well, definitely I need some time to consume those you provided. I will get back to you if I make breakthrough on this direction. Thanks a lot~
  2. L

    Graduate Given Any Measurable Space, Is There Always a Topological Space Generating it?

    Yes, exactly, to prove that \bigcup_{i\in I}{\mathcal{T}_i} is a subset of M or to find out a counterexample puzzles me...
  3. L

    Graduate Given Any Measurable Space, Is There Always a Topological Space Generating it?

    I would guess it is true and have a candidate proof. Let S be the collection of all topological spaces (X,T') such that T' \subset M. Then S is nonempty and partially ordered. If we could prove that every chain in S has an upper bound in S, then we can apply Zorn's lemma to assert the existence...
  4. L

    Graduate Given Any Measurable Space, Is There Always a Topological Space Generating it?

    As well known, for any topological space (X,T), there is a smallest measurable space (X,M) such that T\subset M. We say that (X,M) is generated by (X,T). Right now, I was wondering whether the "reverse" is true: for any measurable space (X,M), there exists a finest topological space (X,T) such...