Locally connected fan

From hyperspacewiki
Jump to: navigation, search

The locally connected fan, written $F_{\omega}$, is the union of countably many straight line segments of length tending to $0$, emanating from a point $p$ and disjoint away from $p$. We can think of $F_{\omega}$ as the dendrite with only one ramification point whose order is $\omega$.

Locally connected fan.gif

Properties

Theorem: $F_{\omega}$ is universal in the class of all $n$-ods.

Proof:

Theorem: Each open image of $F_{\omega}$ is homeomorphic to $F_{\omega}$.

Proof:

Theorem: A confluent mapping defined on $F_{\omega}$ is open if and only if it is light.

Proof: