“Flat” in this context only means that space itself isn’t curved. You can have a sphere in a flat metric space. In fact, the spherical universe you are imagining would be considered “flat.” In this context, the term doesn’t say anything about the number of dimensions.
Imagine a laser that shines two parallels lines away from the earth. You can make a statement about the geometry of space based on the behavior of these two lines.
One of the three will be true: The lines will either intersect, diverge or remain parallel indefinitely. The latter is what is meant by “flat.” Flat just means that space doesn’t fold in on itself or fold outwards.
The universe appears to be *flat* because observations are consistent with two parallel lines remaining parallel for as far out as currently observable.
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