Hello, and thank you for your question.
This could be several things. Please allow me to ask you some questions in order to give you a more informed answer.
Have you had a cold recently? Do you have any itching or mattering of your eyes at all?
The only symptom of whatever I have is the redness in the corner of my right eye. I have no itching, running, soreness, etc. and I have no cold now or anytime recently.
Given the early onset of this redness, it unfortunately can be several things including conjunctivitis (inflammation of the clear skin-like covering of the white of the eye), episcleritis/scleritis (inflammation of the actual white part of the eye) or intraocular inflammation.
As you are completely asymptomatic, however, I suspect this is either an episcleritis or an early viral conjunctivitis (pink-eye is the common vernacular). Episcleritis often just presents as redness without any symptoms and patients end up coming in just because so many people ask them what's wrong, not that they have any symptoms.
I would say if you do not go on to develop discharge, pain, light sensitivity, changes in vision or symptoms such as these or redness in the other eye, then it is okay to just observe this for resolution.
If it is a viral conjuncitivits (pink-eye), then you may begin to have symptoms in the other in a few days. A viral conjunctivitis just has to run it's course; because it is not a bacteria, it dose not respond to antibiotic drops. The key with these is just infection control (frequent handwashing, avoiding touching the eyes/face, avoiding contact with sick or immune compromised persons).
Does this address the questions you have?
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Disclaimer: My opinion is solely informative and does not constitute an informed medical opinion or recommendation. For an informed medical opinion and/or recommendation you must see an eye doctor.
If the condition is conjunctivitis, is the redness usually contained to the current position, or oes it spread to other areas of the same eye or posssibly to the other eye?
If it is a viral conjunctivitis, I would say that would depend on where you work. If you're at a desk job with not a lot of public contact, you could probably go to work and be okay, as long as you used antiseptic hand-gel liberally and washed hands frequently.
But if you are in a food-service, grocery store, hospital, or high public volume place of employment it would be good to stay at home until your eyes looked and felt normal. Before doing that, however, I would go see an ophthalmologist to get examined so you can be sure of that diagnosis. I would hate to have you lose work if you didn't need to.