Question: Lets say you work in Chicago and live in Chicago but put your home address as somewhere in Indiana. Then every week you collect the mail that is being sent to you in Indiana. Wouldn't you save a lot of money assuming the taxes are lower? What prevents people from doing this?

Answer: 
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> What prevents people from doing this?
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* Laws (if you are *lying* about your residency)
* Convenience
* It won't work for income taxes

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> Wouldn't you save a lot of money assuming the taxes are lower?
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According to the [Illinois Department of Revenue](https://www2.illinois.gov/rev/individuals/filingrequirements/Pages/default.aspx):

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> you must file Form IL-1040 and Schedule NR if
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> * you earned enough taxable income from Illinois sources to have a tax liability
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So just *claiming* you live in Indiana won't stop you from owing income tax in Illinois. Plus Indiana might compensate for lower income taxes with higher property taxes, so you might actually may *more* taxes with this scheme.

(I see in the comments you only want to create a PO Box, so this may not apply, but it does apply to other situations where people *legitimately* live across state lines).

NBA players actually have to pay tax to [*all states that they play in*](https://www.slamonline.com/nba/state-taxes-nba-free-agency/). Otherwise, every player would move to a state that has no state income tax.
It's difficult if not impossible to do if you actually live in Chicago, and just have a mail drop in the other state. You'd need to establish effective residency there, by e.g. owning a residence, spending time there, maintaining voter registration, driver's license and so on. All of that is likely to cost more than you'd save in taxes unless you are a) really rich; or b) actually do live in the second state, and just work in the first.

As an example of b, for a couple of years I spent alternate weeks working in California & living in rented lodgings there. The other weeks I'd work from my actual (owned) home in Nevada, where I maintained everything that goes into establishing residency. As a consequence, I only had to pay California nonresident tax on the portion of my income that came from the California client, not on income that I got from my other clients. The critical point here is that I actually WAS a Nevada resident, and had been for many years before taking the California job, so there was no lying about my resident status. It was never questioned, but I could have easily proved it if it had been.

PS: And per comments above, the only reason this actually benefitted me is because only a fraction of my income was from California. Had I had only a single W-2 job in California, I think (though I've never actually worked it out) that I would have paid as much whether I was a resident or non-resident.