My neighbors Persian has infection around the eye?

There is no alternative to a visit to a vet. This can be a condition called conjunctivitis which is an infection which needs an antibiotic eye ointment prescribed by a vet. This condition will cause the eye lids to become red and inflamed but also will cause a thick greenish-yellow discharge from the eyes that will cause the eyes to become stuck shut when the discharge dries and hardens and you would need to gently with a warm wet cotton ball carefully wipe the crust away. Kitty may have allergies or even some type of skin aliment caused by mites or micro-insects. What ever the problem is only a vet can diagnose.

if the problem is getting sharp and they are not paying attention to the cat's health you may inform the police or some one else in charge that they are refusing to bring the pet into a veterinarian. the matter may even go serious.

I understand you do not really want to face your neighbour, but in this case, for the sake of the cat, just go round and let them know.