# 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: