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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 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 an ophthalmic antibiotic ointment at a pharmacy: 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.
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 also 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.