The number of dimensions can change through quantum fluctuations, which were large at the time of the Big Bang but are now small or so rare that for all practical purposes they can be ignored. It is interesting to note in this connection that the number of dimensions does not have to be an integer: it is possible to construct mathematically a space of dimension 1/3 or 9/8, for instance. An object of dimension greater than 1 and smaller than 2 is obtained by taking a planar curve and stretching and folding it innumerable times, but still in such a way that it does not fill entirely any finite region of the plane. An object of this kind is called a fractal curve, and more generally a set of points with a non-integer dimensionality is called a fractal set. According to Linde and other proponents of "chaotic inflationary cosmology", the relatively peaceful universe that nurtures us was born from a fractal and wildly fluctuating "quantum foam": it is a "bubble" of this foam that has greatly expanded in three of the space dimensions and has settled down to nine space dimensions overall.

Mathematical spaces with infinite dimensionality

Spaces with an infinite number of dimensions have been extensively studied by mathematicians and have found an application in quantum theory to represent in an abstract way the infinite possible configurations of a quantum system such as an atom. These abstract spaces of quantum theory were introduced by Hilbert, the great German mathematician who lead the way into 20-th century mathematics [1] .In quantum gravity, each point of physical space has an infinite-dimensional Hilbert space associated with it [2]. That's how complicated the theoretical picture has become, and even stranger ideas are under investigation.

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Notes:

[1] One of Hilbert's achievements is that he obtained the equations of General Relativity a few days before Einstein did, by a different method. You can read about it in Hilbert's online biography. He was a good friend of Minkowski, who was Einstein's math teacher in Zurich and originated the notion of the 'space-time continuum'.
Hilbert's influence was enhanced by the work of his many talented students and assistants at the University of Göttingen, including Emmy Noether. Hilbert's paper on general relativity makes use of Noether's deep insight on the central role of "invariants" in mathematics and their relation to "conserved quantities" in physics. (Electric charge is an example of a conserved quantity.)

[2] A Hilbert space can have a finite or infinite number of dimensions. Finite-dimensional Hilbert spaces correspond to very simple quantum systems: for instance, the spin of the electron, or any other spin-1/2 particle, can be represented in a Hilbert space of 2 dimensions.