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Subspace/induced/relative Topology Definition

  1. Feb 15, 2010 #1
    I'm having trouble understanding the definition of a Subspace/Induced/Relative Topology. The definitions I'm finding either don't define symbols well (at all).

    If I understand correctly the definition is:

    Given:
    -topological space (A,[tex]\tau[/tex])
    -[tex]\tau[/tex]={0,A,u1,u2,...un}
    -subset B[tex]\subset[/tex]A

    The subspace topology on B will be the intersection of B and every part of the topology of A

    OR

    [tex]\tau[/tex]B={0,B,B[tex]\bigcap[/tex]u1,B[tex]\bigcap[/tex]u2,...B[tex]\bigcap[/tex]un}


    ...I apologize in advance for my LATEX work.
     
  2. jcsd
  3. Feb 15, 2010 #2

    Hurkyl

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    That looks correct idea for the subspace topology... but there is a serious problem in your understanding of topology: you seem to be assuming there are only finitely many open sets.
     
  4. Feb 15, 2010 #3
    The u's are finite for illustrative purposes. I wanted to avoid using the form given by Wikipedia because it has insufficient textual explanation.

    They have this definition, but don't specify exactly what a U is:

    [tex]\tau[/tex]B={B[tex]\bigcap[/tex]U|U[tex]\in[/tex][tex]\tau[/tex]}

    It seems that me that you need to say how U fits into the first topology.
     
  5. Feb 15, 2010 #4

    Hurkyl

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    Yes they do: they say a U is an element of [itex]\tau[/itex].

    This is standard set-builder syntax for replacement: on the right of | you introduce a variable and its domain, and on the left of the | you have a function of that variable indicating what should go into the set you're building.
     
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