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?
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).