In set theory, There is a functional dichotomy between a "potential infinity" and a "real infinity". A potential infinity is a value which continues to expand/extend in one way or another indefinitely, Thus constituting a scalar of unlimited magnitude. At T=infinity, Where T=the time elapsed since said value begins expanding, The scalar magnitude of the value would also be infinite. However, T can never equal infinity, Since infinity is greater than any number. As such the value could never reach infinity despite ever approaching it. A real infinity is the exact opposite, Being a value which is infinite by nature. When we ask whether or not the universe is infinity, I expect we mean to ask whether or not it is actually infinite rather than merely potentially. We have already discussed how, Since it is measurable in intervals of time chronologically (that is, All points in time do not occur at the same time), T cannot equal infinity and thus the universe, Spatially or chronologically, Will never truly equal infinity. This is not to say there are no instances in which the concept of infinity and the real world do not intersect. A singularity is by nature and object with infinite density though necessarily finite mass due to its definitional nondimensionality. Beside, The infinity we see in singularities is not the same kind as that we are basing our definition off of as defined in set theory, Simply because it does not constitute a set at all. All real numbers from negative to positive infinity is a real infinite set because it possesses unthinkably multitudinous points, Unlike a singularity, Which is categorically incomparable. Long story short, Real set infinities cannot exist in the universe. As such, The universe can possess instances of infinity, But itself cannot be infinite.
At the Big Bang, It was infinitely dense. Since then it has just been getting less dense as space has expanded. Imagine a large flat rubber sheet with sand placed as closely together as possible on the sheet. And no because the universe had a beginning if the universe was really infinite that's mean no beginning