Some more problems

the game

(1)prove that in a metric space the closure of a countable set has cardinal no at most c.

(2)Let M be a metric space where the closure of every open set is open.Prove that M is discrete.

I am sure these problem are very easy but they are worth a try...

**Posted:** Tue Feb 22, 2005 11:11 am

jmerry

#1: The topology of a metric space is completely detemined by sequences. We have an injection from the closure of $A$ to the set of sequences of elements in $A$ : for each element of the closure, pick a sequence which converges to it.
If $A$ is countable, the set of sequences in $A$ has cardinality no more than that of the real numbers.

**Posted:** Tue Feb 22, 2005 11:17 am

grobber

A sketch for the second one:

Assume there is a point $x$ which has a sequence $(x_n)_n$ converging to it. Now take an open ball around each $x_{2n-1}$ s.t. the closure of the open set formed by the union of these does not contain any of the $x_{2n}$ . The closure of this open set contains $x$ , but does not contain an open ball around $x$ , because it doesn't contain the $x_{2n}$ 's, which yields a contradiction.

**Posted:** Tue Feb 22, 2005 11:30 am

the game

Well if you are here you can certainly here me clapping.Very nice solutions indeed.

**Posted:** Tue Feb 22, 2005 11:35 am