# Dendrite

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:** █