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

$\displaystyle d(x,y)=$ 0 if x=y, 1 if x $\displaystyle \neq$y

Is the trivial topology metrizable?

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- Apr 7th 2009, 11:35 AMAndreametMetric inducing Discrete Topology
Show that the discrete topology on X is induced by the metric

$\displaystyle d(x,y)=$ 0 if x=y, 1 if x $\displaystyle \neq$y

Is the trivial topology metrizable? - Apr 7th 2009, 12:01 PMPlato
- Apr 7th 2009, 01:11 PMAndreamet
- Apr 7th 2009, 01:55 PMPlato
- Apr 7th 2009, 02:30 PMAndreamet
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?

- Apr 7th 2009, 02:48 PMPlato
- Apr 7th 2009, 07:25 PMaliceinwonderland
A discrete topological space is metrizable. You can give each singleton set (an open ball whose radius $\displaystyle r \in (0,1]$ 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.

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The trivial topological space is not metrizable, since it is not a Hausdorff space ( Metrizable spaces are always Hausdorff and paracompact).