# Thread: [SOLVED] Analysis problem

1. ## [SOLVED] Analysis problem

a point metric space X.show {a}^c is open.

2. Originally Posted by oaza
Can someone help me with that proof.
Let x be any point in a metric space X.Prove that {x}^c(complement of x) is open.
I have the same problem to solve and also I do not know how to approach that.

3. Originally Posted by oaza
Can someone help me with that proof.
Originally Posted by oaza
Let x be any point in a metric space X.Prove that {x}^c(complement of x) is open.
Come on you two guys, this is such a fundamental property of metrics.
The distance between two points in a metric space is positive.
$y \in \left\{ x \right\}^c \Rightarrow \quad d(x,y) > 0$
$\left\{ x \right\}^c = \bigcup\limits_{y \in \left\{ x \right\}^c } {B\left( {y;d(x,y)} \right)}$

4. That is really it?!!

5. Originally Posted by Plato
Come on you two guys, this is such a fundamental property of metrics.
The distance between two points in a metric space is positive.
$y \in \left\{ x \right\}^c \Rightarrow \quad d(x,y) > 0$
$\left\{ x \right\}^c = \bigcup\limits_{y \in \left\{ x \right\}^c } {B\left( {y;d(x,y)} \right)}$
I always think proofs are supposed to be so long...lol is it really that simple?

6. Originally Posted by megamet2000
is it really that simple?
That shows that the complement is the union of open balls.