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Thread: semi-metric spaces and qausi-semi developable

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

    prove that every semi-metric space is qausi-semi developabel.
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  2. #2
    Super Member girdav's Avatar
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    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.
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  3. #3
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    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.
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  4. #4
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    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.
    Attached Files Attached Files
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