we only need to show that time is finite in extent, and it follows that space is also finite [...] That does not follow: As our universe is spatially flat (or

we only need to show that time is finite in extent, and it follows that space is also finite [...] That does not follow: As our universe is spatially flat (or close to it), it is consistent with the infinite-space Friedmann model. The cosmic horizon that bounds the observable universe would still exist due to the metric expansion of space.