On the space of Cantor subsets of R3R3

The space of Cantor subsets of R3R3, denoted C(R3)C(R3), is a Polish space. We prove this space is path connected and locally path connected. The group of autohomeomorphisms of R3R3, denoted Aut(R3)Aut(R3), acts on C(R3)C(R3) naturally. This action gives us natural invariant classes of Cantor sets and we show that these classes are in the lower levels of the Borel hierarchy, in fact they are open, closed, FσFσ or GδGδ in C(R3)C(R3). Moreover, we prove that the classification problem of Cantor sets arising from this action is at least as complicated as the classification of countable linear orders.