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Math Help - Metric inducing Discrete Topology

  1. #1
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    Metric inducing Discrete Topology

    Show that the discrete topology on X is induced by the metric

    d(x,y)= 0 if x=y, 1 if x \neqy

    Is the trivial topology metrizable?
    Last edited by Andreamet; April 7th 2009 at 01:12 PM.
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  2. #2
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    Quote Originally Posted by Andreamet View Post
    the metric \color{red}d(x,y)=1 \text{ if } x=y, 0 \text{ if } x \not= y
    That is not a metric! Do you know why?
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    Quote Originally Posted by Plato View Post
    That is not a metric! Do you know why?

    oops, I made a correction (see above)
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    Quote Originally Posted by Andreamet View Post
    oops, I made a correction (see above)
    What are the open sets?
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    Well,every set is open in the discrete topology. I am having trouble connecting the notion of distance to open sets in metrics. I know balls are probably the answer, but how do I build a ball from this metric? epsilon<1.001?
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    Quote Originally Posted by Andreamet View Post
    Well,every set is open in the discrete topology. I am having trouble connecting the notion of distance to open sets in metrics. I know balls are probably the answer, but how do I build a ball from this metric? epsilon<1.001?
    You should have proven that in a metric space each ball is an open set.
    Can you answer the question \left( {\forall x \in X} \right)\left[ {\mathcal{B}\left( {x;0.5} \right) = ?} \right].
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  7. #7
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    Quote Originally Posted by Andreamet View Post
    Show that the discrete topology on X is induced by the metric

    d(x,y)= 0 if x=y, 1 if x \neqy

    Is the trivial topology metrizable?
    A discrete topological space is metrizable. You can give each singleton set (an open ball whose radius r \in (0,1] with respect to the discrete metric d ) in X as a basis element.

    You need to check whether singleton sets form a basis for a discrete topological space X.

    (1) Check if singleton sets cover X.
    (2) If two basis elements have an intersection, then for each x in the intersection, there is another basis element containing x and contained in the intersection.

    For (2), if there is no intersection, (2) is vacuously true.
    -----------------------------------------------------------
    The trivial topological space is not metrizable, since it is not a Hausdorff space ( Metrizable spaces are always Hausdorff and paracompact).
    Last edited by aliceinwonderland; April 8th 2009 at 03:25 PM. Reason: Cor
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