topology proof: Let X = (X, d) be a finite metric space....

[gabyra]

Hi, I need guidance with this proofs. Thanks for your help
1)Let X = (X, d) be a finite metric space, that is, a metric space with a finite
number of points. Prove that every subset of X is open.
Can I take every subset as a singleton and then prove that they are open?, then their union would be open too.
But how do I prove a singleton is an open subset?

Let (an) be a sequence of points in a metric space X that converges to a
point a belongs to X. Let b belongs to X be an arbitrary point. Prove that the sequence of
real numbers
d(an, b), n = 1, 2, 3, . . .
converges in R, and find its limit.
I'm not sure how to go on this one...

 

[pka]

Hi, I need guidance with this proofs. Thanks for your help
1)Let X = (X, d) be a finite metric space, that is, a metric space with a finite
number of points. Prove that every subset of X is open.
Can I take every subset as a singleton and then prove that they are open?, then their union would be open too. But how do I prove a singleton is an open subset?

Because X is finite take \(\displaystyle \frac{{\min \left\{ {d\left( {x_j ,x_k } \right):j \ne k} \right\}}}{2} = \delta\).
Now you know that \(\displaystyle \delta > 0\) because the points are distinct.
Each ball \(\displaystyle B\left( {x_j ;\delta } \right)\) is an open set. So singleton sets are open.
Every subset of X is the union of finite collection of open sets.

Let (an) be a sequence of points in a metric space X that converges to a point a belongs to X. Let b belongs to X be an arbitrary point. Prove that the sequence of real numbers d(an, b), n = 1, 2, 3, . . .converges in R, and find its limit.

For any points \(\displaystyle \left\{ {a,b,x} \right\} \subseteq X\) then \(\displaystyle \left|{d(a,x)-d(b,x)}\right|\le d(a,b)\).
Use that and replace.