In the Glossary of his 1997 book, THE INFLATIONARY UNIVERSE, Alan Guth has this to say:
OK so far, even though I thought 'open' and 'closed' was usually referring the spatial geometry. anyway, here is the part of interest:OPEN UNIVERSE: A homogeneous and isotropic universe is said to be temporally open if gravity is not strong enough to eventually reverse the expansion, so the universde goes on expanding forever. It is said to be spatially open...if triangles contain less than 180 degrees....
So I checked to see what he said about a 'closed universe' and sure enough it is consistent:...IF EINSTEIN'S COSMOLOGICAL CONSTANT IS ZERO, AS IS FREQUENTLY ASSUMED, THEN A UNIVERSE WHICH IS TEMPORALLY OPEN IS ALSO SPATIALLY OPEN, AND VICE VERSE.
Is it obvious that open and closed time and space go together? Why only for a zero cosmological constant? And what are the time and space relationships if the cosmological constant isn't exactly zero, say small positive, as is currently thought?....IF EINSTEIN'S COSMOLOGICAL CONSTANT IS ZERO, AS IS FREQUENTLY ASSUMED, THEN A UNIVERSE WHICH IS TEMPORALLY CLOSED IS ALSO SPATIALLY CLOSED, AND VICE VERSA.