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Math Help - Hausdorf space and discrete topology

  1. #1
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    Hausdorf space and discrete topology

    What is the difference between a hausdorf space and a discrete topological space?
    I know any two points of a hausdorf space have disjoint neighbourhoods- would that not mean that the space be a discrete topology?

    I have read that hausdorf can be made more fine but would that not mean the more coarse space could have elements that do not have disjoint neighbourhoods?

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  2. #2
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    Quote Originally Posted by Chad950 View Post
    What is the difference between a hausdorf space and a discrete topological space?
    I know any two points of a hausdorf space have disjoint neighbourhoods- would that not mean that the space be a discrete topology?
    I have read that hausdorf can be made more fine but would that not mean the more coarse space could have elements that do not have disjoint neighbourhoods?
    I think that you need to study the definition of Discrete_topology
    In particular, notice that each singleton set is open.
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  3. #3
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    Oh I see!
    Is there any other discrete space that is not hausdorf (other than that based on the singleton set)?
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  4. #4
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    The point is that a "discrete topological space", by definition, has every singleton set open (and therefore, every set is open. There is no "other" discrete space.
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  5. #5
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    Every discrete space is hausdorff, but not every hausdorff space is discrete.
    Take for example the set of real numbers with the natural topology, i.e. the topology which has all open intervals as a basis. This topology is Hausdorff, because for any two different points x<y \in \mathbb{R} the open intervals (x-1,\frac{x+y}{2}) and (\frac{x+y}{2},y+1) are disjoint open neighbourhoods of x resp. y.
    But it is not discrete, because the singletons are not open.
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