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Topological properties on Linear spaces

  1. Mar 18, 2009 #1
    Is it reasonable to work with a linear space whose subspaces are considered as open subsets of the linear space when the linear space is considered as a topological Space? Actually, this linear space is spanned by a topological space with known topology.
  2. jcsd
  3. Mar 18, 2009 #2
    The subspaces of a vector space do not form a topology because they do not satisfy the axioms of a topological space: The empty set is not a subspace (subspaces have to contain 0) and the union of subspaces is not a subspace in general.

    You may want to read http://en.wikipedia.org/wiki/Topological_vector_space" [Broken] which describes the topologies one usually considers on a vector space.

    Somewhat related to your idea is the http://en.wikipedia.org/wiki/Zariski_topology" [Broken], where the closed sets are the zero sets of a system of polynomials (vector subspaces are a special case).
    Last edited by a moderator: May 4, 2017
  4. Mar 18, 2009 #3
    can't we make a linear space a topological space by construction, since we know that every non empty set can be made a topological space. ofcourse a linear space can never be empty for it contain the zero vector. Please still consider my question above, because i am really working on it
  5. Mar 18, 2009 #4
    There are many ways in which a vector space can be equiped with a topology, I was just saying that your particular choice of open sets would not work. Of course one can choose a topology where the subspaces are open sets, for example the topology of all subsets, but this is not a very interesting topology.
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