let $D$ be the decopomsition of the plane into concentric circles about the origin.
i need to demonstrate that $D$ is homeomorphic to $A$={$x\in \mathbb R$ :$x\geq$ 0} and $D$ is upper semicontinuous.
this is what i did to demonstrate.
let $P$ the natural map ($P:\mathbb R^2\longrightarrow D$) by letting $P(x)$, for $x\in\mathbb R^2$ be the element of $D$ containing $x$
note: the topology the descomposition sapace $D$ is the quotient topology induced by $P$
let $f:\mathbb R^2\longrightarrow A $, $f(x,y)=x²+y²$. (we notice that $f$ is surjective), then we endow $A$ with the quotient topology induced by $f$.
so we have to , there is a homeomorphism $H:A\longrightarrow D$ , and it is also fulfilled $P= H\circ f$ (*).
To prove that $D$ is upper semicontinuous.this is what i did to demonstrate.
consider the theorem:
the natural map $P$ associated with a decomposition space $D$ of $X$ is closed iff $D$ is upper semicontinuous.
$P= H\circ f$ is closed because $H ,f$ are map closed
the theorem used in(*) is as follows:
if $Y$ has the quotient topology induced by $f:X\longrightarrow Y$, then $Y$ is homeomorphic to the descomposition space $D$ whose element are the sets $f^{-1}(y), y\in Y $ , under a homeomorphism $H:Y\longrightarrow D$ such that $P= H\circ f$.
is my reasoning okay?.
any suggestion or contribution is welcome!!
If someone knows how to demonstrate it in another way, I would appreciate it.
the definition of upper semicontinuous ,that is known is:
An open set $V$ in a topological space $X$ is saturated relative to a give decomposition $D$ of $X$ iff $V$ is a union of elements of $D$ (i.e iff $V=P^{-1}(W)$ for some open set $W$ in $D$). A decomposition $D$ is upper semicontinuous iff for each $F\in D$ and each open set $U$ in $X$ containing $F$, there is some saturated open set $V$ in $X$ with $F\subset V\subset U$