As far i know, there are a few reasons which can give rise to a red dot in your eyes. Generally speaking, it can stem from the eye infection which leads to your situation. So for your situation, you should not touch your eyes with your dirty hand. And then, just have some cool teabag under your eyes. And it can be effective. On the other hand, it can be the broken blood vessel because there are a multiple of blood vessels in our eyes as well. If they are broken, our eyes can appear a red dot. You should take it easy because it usually go away on its own under such circumstances after a short period of time enough rest and a balanced diet. Certainly, it can also be a symptom of pink eye. But if you suffer from pink eye, you would feel itching. Therefore, I think you can observe your situation and if the situation become worse, you'd better have a physical examination as soon as possible.