# Peano continuum

A Peano continuum is a continuum that is locally connected at each point.

# Properties

**Theorem:** Let $X$ be a Peano continuum. Then for every $\epsilon > 0$, $X$ is the union of finitely many Peano continua whose diameter is less than $\epsilon$.

**Proof:** █

**Theorem:** Every Peano continuum is arcwise connected.

**Proof:** █

**Theorem:** Every Peano continuum has a convex metric.

**Proof:** █

**Theorem:** If $X$ and $Y$ are Peano continua with $X \cap Y \neq \emptyset$, then $X \cup Y$ is a Peano continuum.

**Proof:** █

**Theorem (Wojdyslawski’s Theorem):** If $X$ is a Peano continuum, then the hyperspaces $2^X$ and $C(X)$ are absolute retracts.

**Proof:** █

Characterization of when 2^X is homogeneous

Characterization of when C(X) is homogeneous

All open subsets of Peano continuum are locally arcwise connected

Connected subset of Peano continuum is arcwise connected

Separable locally compact Peano space is arcwise connected

Peano implies contractible hyperspaces

Graphs are Peano continua