Bendability

 18 September 05:24   A amplitude is bunched if every accessible awning admits a bound subcover. This can be formalized as follows:

    A topological amplitude $X$ is bunched , if for every ancestors $\_$ of accessible sets with there exists a bound set $Fsubseteq\; I$ such that .

    Compactness can aswell be bidding by one of the afterward agnate characterisations:

    Theorem Every bunched subset of a Hausdorff amplitude is closed.


    Proof: Let $K$ be compact. If $K^c$ is empty, then $K$ is the aforementioned as the space; appropriately closed. Accept not; that is, there is a point $x\; in\; K^c$. Then for anniversary $y\; in\; K$, by the Hausdorff break adage we can acquisition $U\_y$ and $V\_y$ disjoin, accessible and such that $x\; in\; U\_y$ and $y\; in\; V\_y$. Back $K$ is bunched and the accumulating $$ covers $K$, we can acquisition a bound amount of credibility $y\_1,\; y\_2,\; ...\; y\_n$ in $K$ such that:

    :$K\; subset\; V\_\; cup\; V\_\; ...\; V\_$

    It then follows that:

    :$x\; subset\; U\_\; cap\; U\_\; ...\; U\_\; subset\; K^c$. $square$