# semi-metric spaces and qausi-semi developable

• Jul 12th 2011, 02:37 AM
student2011
semi-metric spaces and qausi-semi developable
prove that every semi-metric space is qausi-semi developabel.
• Jul 12th 2011, 03:20 AM
girdav
Re: semi-metric spaces and qausi-semi developable
Can you write the definitions of "semi-metric space" and "quasi-semi developable"? It's the first step to solve the problem.
• Jul 12th 2011, 10:54 AM
student2011
Re: semi-metric spaces and qausi-semi developable
I know the definition of both semi-metric spaces and qausi-semi developable spaces. A space X is called a semi-metric space if there is a distance function $\displaystyle d:X\times X\to R$ such that:
1) $\displaystyle d(x,y)=d(y,x)>=0$
2) $\displaystyle d(x,y)=0$ iff $\displaystyle x=y$
3) $\displaystyle d(x,A)=0$ iff $\displaystyle x$ is a limit point of $\displaystyle A$

A topological space $\displaystyle X$ is said to be a qausi-semi developable if there exist a sequence $\displaystyle G=G_n$of subsets of $\displaystyle X$ such that for each $\displaystyle x\in X$ and each open set $\displaystyle U$ containing $\displaystyle x$there exists an n with $\displaystyle st(x,G_n)$ is contained in $\displaystyle U$.

So in order to prove that every semi-metric space is a qausi-semi developable space, we should get the a qausi-semi development. How can I get it, please guide me.
• Jul 18th 2011, 01:38 AM
student2011
Re: semi-metric spaces and qausi-semi developable
Ok, thank you very much, I got the answer, in fact every semimetrizable spaces are semi-developable and thus qausi-semi developable.

I attached the answer in the attachement.