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I'm sorry to hear about your cat‘s eye. If attempting to treat at home you want to be very careful first of all that the eye continually looks better through treatment, and no signs of rubbing at it, reddening worse, or cloudiness appear in the eye.
You cannot use Vigamox safely, as the pH os formulated for the human eye, and can cause more damage than good in treating your kitty's eye.
You should rinse the eye with sterile saline, like that found in regular contact lens solution, (not the ones with disinfectant but just normal saline/ NaCl) and do not let the tip of the container touch the eye. You can wipe around the eye with the saline soaked on a soft cotton ball or gauze, but don't use anything harsh like paper towels.
You can also find a pet safe ophthalmic antibiotic ointment at most pet supply stores, catalogues, pet stores and some pharmacies: like terramycin, triple antibiotic (neo/poly/bac), etc. Ask the sales people if you need help, but it does need to be safe for the eye, not a topical one for skin. After rinsing the eye, apply the ointment 3 times a day (every 8 hrs) for 10 to 14 days.
It can also be bought online, here is a link of just one sight that sells it, if interested:
This is likely conjunctivitis, but could also be a scratch or ulcer forming and if not starting to clear up in a couple of days, you'll need a vet to check it. It is very possible some debris or litter settled in your cat's eye, causing this problem.
Best of luck, and let me know if I can help further.