Rational continua

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A continuum $X$ is said to be rational provided that every point $p \in X$ has a local basis $B_p$ such that the boundary of each member of $B_p$ is at most countable.

Properties

Proposition: The property of being a rational continuum is not a Whitney property.

Proof:

Proposition: The property of not being a rational continuum is not a Whitney property.

Proof: