To contract a "cold" or other virus, an animal would have to have contact with another infected animal. True there are some viruses that cause repeated episodes of local (in and around the eye in this case) inflammation and increased discharge with no history of recent cat to cat contact, we might expect other symptoms in the case of viral diseases (2 eyes affected, sneezing, coughing, nasal discharge, fever, decreased appetite, etc). We have none of that. I am skeptical that a "cold" exists in this case.
It is more likely that:
1. your cat may have an obstructed tear duct, a usually benign situation that self resolves. Mucus portion of tears (they also have an aqueous portion) accumulates on eye corner.
2. your cat could have minor irritation in one eye from floor particulates such as dust
3. your cat could have a more serous irritation, and perhaps a corneal ulcer from a scratch to the cornea (clear part of eye) that will require a doctors care.
See your doctor ASAP to determine what the actual problem is and begin treatment if needed. Eye problems can rapidly get worse and become more difficult to resolve as well as more expensive to treat. No time to waste!
Arnold L. Goldman DVM, MS
MedHelp & PDOC