I'm following this tutorial: https://www.youtube.com/watch?v=jCVEtLjpeB8
And when I add the Subdivision Surface modifier there's a hole at the bottom of my cup. I have no idea why this happens and how to fix it.
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The problem is that you haven't enabled the Merge option in the Screw modifier. Therefore you have many overlapping vertices in the center area. This creates a bad shading since you have set the object to Shade Smooth but didn't enable the Normals > Auto Smooth option in the Object Data Properties.
Auto Smooth will get rid of the visual effect, but you still have a lot of overlapping vertices, so you should definitely enable the Merge option, otherwise your mesh will stay unconnected in the center. By the way, the inner vertex of the glass bottom is not exactly in the center, so there is a hole. I would select it and set the Y value to 0.
The slightly off vertex: