Metric inducing Discrete Topology

• Apr 7th 2009, 11:35 AM
Andreamet
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 $\neq$y

Is the trivial topology metrizable?
• Apr 7th 2009, 12:01 PM
Plato
Quote:

Originally Posted by Andreamet
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?
• Apr 7th 2009, 01:11 PM
Andreamet
Quote:

Originally Posted by Plato
That is not a metric! Do you know why?

oops, I made a correction (see above)
• Apr 7th 2009, 01:55 PM
Plato
Quote:

Originally Posted by Andreamet
oops, I made a correction (see above)

What are the open sets?
• Apr 7th 2009, 02:30 PM
Andreamet
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 PM
Plato
Quote:

Originally Posted by Andreamet
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 $\left( {\forall x \in X} \right)\left[ {\mathcal{B}\left( {x;0.5} \right) = ?} \right]$.
• Apr 7th 2009, 07:25 PM
aliceinwonderland
Quote:

Originally Posted by Andreamet
Show that the discrete topology on X is induced by the metric

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

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.
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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).