Peano continuum

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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

See Also

Peano space