Assuming the Continuum Hypothesis, there is a compact, first countable, connected space of weight $\aleph_1$ with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.

Keywords:

connected space, Continuum Hypothesis, proper forcing, irreducible map connected space, Continuum Hypothesis, proper forcing, irreducible map

MSC Classifications:

54D05, 03E35 show english descriptions Connected and locally connected spaces (general aspects)
Consistency and independence results 54D05 - Connected and locally connected spaces (general aspects)
03E35 - Consistency and independence results

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