Dendrite

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A dendrite is a locally connected dendroid.

Properties

Theorem: (Characterization of dendrite) A locally connected continuum $X$ is a dendrite if and only if $X$ contains no simple closed curve.

Proof:

Theorem: Let $X$ be a dendrite. Then every subcontinuum of $X$ is a dendrite.

Proof:

Theorem: Let $X$ be a dendrite. If $Y$ is a subcontinuum of $X$ and $C$ is a connected component of $X \setminus Y$, then $C$ is open in $X$ and $\overline{C} \setminus C$ has exactly one point.

Proof:

Theorem: Let $X$ be a dendrite, then $X$ is hereditarily unicoherent.

Proof:

Theorem: Every connected subset of a dendrite is arcwise connected.

Proof:

Theorem: Let $X$ be a dendrite. The order of $p \in X$ is equal to the number of components of $X \setminus \{p\}$.

Proof:

Theorem: The set of all end points of a dendrite is $0$-dimensional.

Proof:

Theorem: The set of all endpoints of a dendrite is a $G_{\delta}$-set.

Proof:

Theorem: The set of all points of order $2$ (ordinary points) of a dendrite $X$ is dense in $X$.

Proof:

Theorem: Every dendrite has at most countably many ramification points.

Proof:

Theorem: No dendrite contains points of order $\aleph_{0}$ or $\mathfrak{c}$.

Proof:

Theorem: In any dendrite the set of all its end points is dense if and only if the set of all its ramification points is dense.

Proof:

Theorem: Dendrites are absolute retracts.

Proof:

Theorem: A continuum $X$ is a dendrite if and only if for every compact space $Y$ and for every light confluent mapping $f \colon Y \rightarrow f(Y)$ such that $X \subset f(Y)$ there is a copy $X'$ of $X$ in $Y$ for which the restriction $f|X' \colon X' \rightarrow X$ is a homeomorphism.

Proof:

Examples

  1. Locally connected fan
  2. Locally connected comb