My girlfriend once got an infected eyelid from a cat scratch. She went to the local eye hospital who took it VERY seriously, did a thorough examination and gave her a heavy dose of antibiotics on the spot (injected, I think).

You might think they were worried about infection spreading to the eyeball and maybe causing blindness. They weren't.

The real risk they had in mind was meningitis. Infections like that can spread up the optic nerve and into the brain within hours.