# Thread: semi-metric spaces and qausi-semi developable

1. ## semi-metric spaces and qausi-semi developable

prove that every semi-metric space is qausi-semi developabel.

2. ## 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.

3. ## 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.

4. ## 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.