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Topologies Finer vs. Coarser

  1. Feb 10, 2009 #1
    If T and T' are topologies on X and T' is strictly finer than T, what can you say about the corresponding subspace topologies on the subset Y of X?

    See the thing is... I know that they CAN be finer but that's might not be for every case. because Y is JUST a subset of X. This does not necessarily mean finer or coarser.

    How do i put this in terms of symbols in my proof?
     
  2. jcsd
  3. Feb 10, 2009 #2
    I would reread the definition of subspace topology.

    For your question, suppose A is open in the subspace topology induced by T. Then by definition, there exists U open in X such that A = U int Y. Since U is also open in the T' topology this shows that A is open in the topology induced by T'. This shows that the T' induced topology is finer than the T-induced topology.

    But it needn't be strictly finer (Hint: let Y be a single point).
     
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