Furthermore, if you own a ∞-dimensional house, and your house is wide enough to accomodate 2 rooms along each wall (a pretty reasonable size IMO!), then you have more than enough space to run a

Hilbert Hotel. (In fact,

much more than that. But we'll get to that.) Here's how you do it: suppose again that your house is in the shape of a cube, between <0,0,0,...> and <1,1,1...>. Divide the interior of the house into rooms of dimensions <1,½,½,½,...> (remember, the first coordinate is the vertical; we want at least adequate head room in each room so that our guests don't have to crawl -- that would not be very nice). How many of these rooms can we fit into our house?

A

lot. For example, we can fit a room between <0,0,0,...> and <1,½,½,½...>, and another room between <0,½,0,0,0...> and <1,1,½,½,½,...>. And a room between <0,0,½,0,0,...> and <1,½,1,½,½,½,...>. And another room between <0,0,0,½,0,0...> and <1,½,½,1,½,½,...>. For any natural number n, we can fit a room between <0, ... {n zeros}, ½, 0, 0, ....> and <1, ... {(n-1) ½'s}, 1, ½,½,½,...>. So we have at least 1 room per natural number, and so we rent them out and turn our house into a Hilbert Hotel.

But actually, we can run a hotel

much bigger than a Hilbert Hotel, all within the same house! To see this, let's return to a finite N-dimensional space and look at our floor plan. The floor is basically an (N-1)-cube with 2

^{N-1} vertices. Since we can fit 2 rooms per wall, that means we can fit 1 room per vertex of the floor plan. (We're simply subdividing our floor in half along each axis.) Now let N diverge to ∞. Since the floor of our ∞-dimensional house is still an ∞-cube, that means we can fit as many rooms in our house as there are vertices in the ∞-cube, which is an uncountable number. So not only we can run a single Hilbert Hotel; we can even run an infinite number of them all within our humble little house!

And in fact, much more than that. A countably infinite number of Hilbert Hotels still would not fill up all our rooms. We can run an

uncountable version of the Hilbert Hotel -- let's call it the Cantor Hotel, where we can accomodate an uncountable number of guests!!

By comparison, the set of guests in a Hilbert Hotel has measure zero compared to the number of guests in our Cantor Hotel, meaning that our hotel is basically still empty after receiving all the guests from the nearby Hilbert Hotel when they closed down because of a nasty water leak that made all their rooms unoccupiable. We can take in the guests from

all the Hilbert Hotels in our town, and we'd still be able to accomodate all of them -- assuming there are a countable number of Hilbert Hotels in town.

Of course, this same argument can be applied to

any ∞-dimensional house that's big enough to be subdivided into smaller rooms, so essentially

every house in town can run a Hilbert Hotel.

But by the same argument, if our town consists of a map with blocks of N×N×N×... houses, for some small number N>1, that means there are an uncountable number of houses in town, and therefore an uncountable population of townsfolk. So a Hilbert Hotel wouldn't be able to accomodate the entire population of the town, but we don't really expect the number of visitors to match the number of local townsfolk anyway, so we may reasonably assume that we'd only get a countable number of visitors, and they can all stay at the local Hilbert Hotel. But in the event that a disaster destroys the next town over, there'd be an influx of an uncountable number of fleeing refugees. In which case we just have to designate one Safe House -- which has an uncountable number of rooms -- to accomodate them all, without upsetting the rest of the local townsfolk. No problem.