Is it a emergency when redness all of a suddenly appears in one of your eyes?

it could be acute closed angle glaucoma ,

which is a medical emergency because you could lose sight in that eye .

usually it is accompanied by

-tummy pain

-eye pain, the eye ball is hard like a rock.

EMERGENCY - no.

Could something be wrong, yes.

It could be pink eye - if that's the case, you will also have pain and probably itching, with some swelling, along with your eye being "glued" shut when you wake up in the morning with a bunch of yellow gunk. If this is the case, you need to go to the doctor and get a prescription because pink eye is contagious.

If you by chance somehow broke a blood vessel in your eye (which is unbelievably common) it won't just be kind of pink or slightly red, it will be bright red and very scary looking, but there is really nothing that can be done about it - it takes about two weeks for it to eventually go away on it's own.

If it's neither of the above two, I wouldn't be alarmed about it at all as long as there is no pain or discomfort associated with it, maybe you just got something in your eye that irritated it or something. If that's the case, give it a couple of days and see if it goes away. It probably will.

Missy Chel... is correct.

Notice if there is drainage from the eye or if your eye makes crusties and is possibly stuck together in the morning. . first symptom of pink eye which will ONLY go away with the use of a prescribed medicine from an Opthalmologist.

Yes,you can go see your Optometrist but by lisencing they are only capable of prescribing certain kinds of medicines and the medicines that are needed to treat or cure "pink eye' or Conjunctivitis can usually ONLY be prescribed by an Opthalmologist.(Few family physicians know exactlly what antibiotic is needed and general prescribe medicine that is pretty generic and rarely treat the problem properly.)

Broken vessels can come from straining either while sneezing,lifting heavy objects,using the bathroom for #2 ,could come from rubbing your eye which would also be another symptom of "pink eye". . scratchiness or you could be showing signs of high blood pressure. . one of the first symptoms of people who may not even know they have high blood pressure.

It could be caused by infection but usually caused by straining yourself too much. Have you lifted a heavy object recently or had a problem with constipation? These can cause tiny blood vessels to break in your eye and cause the redness. It will go away in a short time.

could be pink eye, an infection or it could be a broken blood vessle and that can be serious.