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What does the term "topology" mean in geological sciences? Is there an agreed upon distinction between the two terms "topography" and "topology"?

When trying to answer this question, I came across the article "On the topology of topography: a review" by Keith Clarke and Boleslo E. Romero, however from reading the abstract of this article it seems like they use the mathematical definition of topology. On the other hand, in this answer, the term "topology" seems to be used as a synonym to topography.

user2552936
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  • Topology - topographic study of a particular place specifically : the history of a region as indicated by its topography. For topography - the configuration of a surface including its relief and the position. – Fred Feb 05 '21 at 20:32
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    @Fred - That is a misuse of the term. Topology and topography are distinct concepts. – David Hammen Feb 05 '21 at 22:10

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Topology is a study of deformable shapes and connectivity. Topography is a study of more or less non-deformable shapes. A coffee cup that has an intact handle and a donut with a hole in the middle are equivalent shapes topologically, but obviously are not equivalent shapes topographically.

David Hammen
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    Cartography and GIS use a specialized meaning of "topology" which involves spatial relationships and connectivity between surface features. If you ever worked with an ArcGIS coverage, then you've encountered this notion. – Spencer Feb 06 '21 at 18:41
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    @Spencer That's mostly to ensure that, for example, roads that are known to be contiguous are marked as such. Relying solely on topographic clues such as GPS measurements would result in maps that have lots of dead-ends when in fact the road continues on. Ensuring that roads don't dead-end on a map at various measurement boundaries when in fact they are not dead-end roads at those boundaries uses topology as topology addresses connectivity issues, which I noted in my answer. – David Hammen Feb 06 '21 at 18:47
  • Topology studies related ideas such as homology and homotopy. In the case of roads represented by finite sets of edges and vertices, i.e. a graph, adjacency-preserving maps give a discrete flavour of homeomorphism. – Galen Aug 21 '22 at 17:55