The universe has a four-dimensional topology

 

I am told the term topology is incorrect in this context and the correct usage would be, "The universe has a four-dimensional topography".

 

Which is correct and why? What are the differences?

 

Feedback is appreciated, thanks.

Topography is local information, in this context we mean local geometry. That is the local shape of the universe.

 

 

Topology is global information, in particular the global shape of the universe.

 

Cosmology theories that come from general relativity can really only describe the local geometry of the universe. Einstein's field equations inherently describe local geometry and not the global topology of a space-time.