Hotel Infinity:  Part Three

The third part of Hilbert's Hotel Paradox, on infinite sets of infinite sets.  

As the proprietor of an infinitely large hotel, you pride yourself on welcoming everyone, even when the rooms are full.  Your hotel becomes very popular among infinite sports teams, as a result.  

 

Recruitment season presents a challenge, however, when many infinite teams arrive at once.  How many infinite teams can stay in a single infinite hotel?  

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