Define a meteric on a X so the associated topology is trivial

**rqeeb** · August 27th 2011, 12:07 AM

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.!

**FernandoRevilla** · August 27th 2011, 01:49 AM

Originally Posted by rqeeb

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

Right, for every $x\in X$ we have $B(x,1/2)=\{x\}$ which implies $\{x\}$ is open etc.

**rqeeb** · August 27th 2011, 02:23 AM

Originally Posted by FernandoRevilla

Right, for every $x\in X$ we have $B(x,1/2)=\{x\}$ which implies $\{x\}$ is open etc.

thank u sir

**rqeeb** · August 27th 2011, 02:24 AM

can any one help me in the 2nd part

**FernandoRevilla** · August 27th 2011, 02:58 AM

Originally Posted by rqeeb

can any one help me in the 2nd part

Suppose $X$ has at least two different elements $x$ and $y$ and denote $d=d(x,y)>0$ . Then, $B(x,d/2)$ is a nonempty open set and $y\not\in B(x,d/2)$ . We can conclude.

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