Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The definition of dimension

  1. Nov 28, 2007 #1
    What is the exact definition of the dimension of a topological space?
     
    Last edited: Nov 29, 2007
  2. jcsd
  3. Nov 28, 2007 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    ?? Your title is "the definition of dimension" but your question is "what is the exact definition of a topological space?" Which is it? The definition of "dimension" depends strongly on exactly what kind of space you are dealing with. The definition of "topological space", however, is quite simple:

    A topological space is a set with a topology!

    And a topology (for set X) is a collection, T, of subsets of X satisfying:
    The empty set is in T.
    The entire set X is in T.
    The union of any collection of sets in T is also in T.
    The intersection of any finite collection of sets in T is also in T.

    For any set X, whatsoever, the following are topologies on T:

    The collection of all subsets of X. (Often called the "discrete" topology.)

    The collection containing only the empty set and X. (Often called the "indiscrete" topology.)
     
  4. Nov 29, 2007 #3
    I made a mistake. What I want to ask is "the definition of the dimension of a topological space"
     
  5. Nov 29, 2007 #4
    There are many notions of dimension, as HallsofIvy warned you. I think the most general one is the topological dimension, also known as the covering dimension.
     
  6. Nov 29, 2007 #5

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

  7. Nov 29, 2007 #6
    Those ones require a metric though. A non-metrizable space should still have a topological dimension.

    Other useful dimensions are upper and lower Minkowski dimensions (related to the box-counting dimension) and the Assouad dimension (aka Bouligand dimension).

    It's interesting to note that the topological dimension of a space is also equal to the infimum the Hausdorff dimensions of all spaces to which it is homeomorphic, in the case when your space is a separable metric space (so that this makes sense). I think it's pretty cool that those two ways of defining topological dimension give the same number! (I have no idea how to prove it. I don't think it's easy.)
     
  8. Nov 29, 2007 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    cohomological dimension is another cool definition.
     
  9. Nov 29, 2007 #8

    Chris Hillman

    User Avatar
    Science Advisor

    Ditto Xevarion; clearly the OP wants the Lebesgue covering dimension. Many good "general topology" textbooks cover this--- er, no pun intended :rolleyes:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: The definition of dimension
  1. 4 dimensions (Replies: 47)

  2. Dimension Question. (Replies: 1)

  3. Dimension of a set (Replies: 6)

  4. 7 dimensions (Replies: 1)

  5. Negative dimension? (Replies: 2)

Loading...