In one sense it does, and in another it doesnt.
In a closed universe, space is always bounded in the sense that it has a finite volume at any epoch. In an infinite universe, space is always unbounded, even at the Big BAng itself. However, in terms of the histories of particles, the Big Bang is the origin of all world lines since along any world line the past history of a particle ends abruptly at the Big Bang.
In this sense, even in an infinite universe, which by the way has always had an infinite spatial volume even at the Big Bang, the Big Bang 13.7 billion years ago does provide a boundary to the universe. But this boundary exists in space-time, not simply in the 3-dimensional spatial portion of its geometry. Our universe had no 'boundary' in space because the evidence shows we live in an infinite universe, whose space would have been infinite even at the Big Bang. However, it did have a boundary in time because it started 13.7 billion years ago and time did not exist before then according to the current theory.
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