# Thread: Define a meteric on a X so the associated topology is trivial

1. ## Define a meteric on a X so the associated topology is trivial

What is the topology determined by the meteric on X given by d(x,y)=1 for x doesnt equal y and d(x,y)=0 for x=y?

for me it looks the discrete topology. Is it right?

can we define a meteric on a given set X so that the associated meteric topology is trivial.!

2. ## Re: Define a meteric on a X so the associated topology is trivial

Originally Posted by rqeeb
for me it looks the discrete topology. Is it right?
Right, for every $\displaystyle x\in X$ we have $\displaystyle B(x,1/2)=\{x\}$ which implies $\displaystyle \{x\}$ is open etc.

3. ## Re: Define a meteric on a X so the associated topology is trivial

Originally Posted by FernandoRevilla
Right, for every $\displaystyle x\in X$ we have $\displaystyle B(x,1/2)=\{x\}$ which implies $\displaystyle \{x\}$ is open etc.
thank u sir

4. ## Re: Define a meteric on a X so the associated topology is trivial

can any one help me in the 2nd part

5. ## Re: Define a meteric on a X so the associated topology is trivial

Originally Posted by rqeeb
can any one help me in the 2nd part
Suppose $\displaystyle X$ has at least two different elements $\displaystyle x$ and $\displaystyle y$ and denote $\displaystyle d=d(x,y)>0$. Then, $\displaystyle B(x,d/2)$ is a nonempty open set and $\displaystyle y\not\in B(x,d/2)$ . We can conclude.