Standard universal dendrite

From hyperspacewiki
Jump to: navigation, search

A standard universal dendrite of order $m$ is a dendrite $D_m$ with the property that each ramification point of $D_m$ is of order $m$ and for each arc $A \subset D_m$, the set of ramification points of $D_m$ which belong to $A$ is dense in $A$.

Standard universal dendrite '"`UNIQ-MathJax10-QINU`"'.Standard universal dendrite '"`UNIQ-MathJax12-QINU`"'.Standard universal dendrite '"`UNIQ-MathJax13-QINU`"'

Properties

Theorem: For each $m \in \{3,4,\ldots,\omega\}$, $D_m$ is universal in the class of all dendrites for which the order of their ramification points is less than or equal to $m$.

Proof:

Theorem: If $m, n \in \mathbb{N}$ with $3 \leq m < n$, then there exists an open mapping of $D_n$ onto $D_m$.

Proof:

Theorem: For each $m \in \{3,4,\ldots,\omega\}$ a monotone surjection of $D_m$ onto itself is a near homeomorphism if and only if $m=3$.

Proof: