It depends upon what you mean by space. Metaphysics after all means thinking about the basic constituents of what is physical: space, time, matter etc. It is about what is necessarily the case. However, such thinking often finds a place for what is not the case, because we can ask why is this not the case.
For example, space is 3d. But we have established consistent descriptions for a geometry of any dimension. So why is it space is not 4d or 5d or higher but actually 3d?
This has turned out to be a very good question. And there may be very good reasons for it to be 3d. We just don't know yet.
Until recently, no physical theory determined the dimensionality of space. It was taken as an empirical given. It's a physical constant that is not usually taken to be one.
One clue, however, is that string theory determines the space dimension to be 25d (+1 of time). Of course it would be much nicer if it was the value we know, 3d. It may be that other ideas can bring it down. In fact, one does, supersymmetry. In that case string theory says space must be 9d (+1 of time). But of course, the jury is still out on whether supersymmetry is realised in our universe.
Now, there are many kinds of higher dimensional spaces. No mathematician actually visualises these. What they do is invent and discover tools that help them work these spaces. When they imagine spaces, it is the low-dimensional spaces that they imagine: 1, 2 & 3d.
This is one area where popular science books fall down on. They don't make this clear, instead relying on visualisations. For example, one tool we have for building spaces is by multiplying them: a line segment multiplied by another one gives a square. A line segment multiplied by a circle gives a cylinder. Whereas a circle times a circle is a torus. We can also add them, a circle plus another circle - is, drum roll, just two circles!
Consider an analogy: Since mass education became widespread, most people can add 25,667,778 to 3,445,556 but no-one actually imagines either of these two numbers. What they do is use an algorithm taught at school. However, ask them to add 2 to 3, and then they can easily imagine these two numbers and they can imagine - that is directly visualise - adding them together too. Moreover, the properties we can establish here also carry on for much bigger numbers. This shows the utility of thinking about 'small' cases.