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October 25th, 2016, 08:38 PM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 270 Thanks: 23 Math Focus: Number theory  A "topology" preserving edges and vertices
Is there a topology, for a dimension D figure, that preserves its d<D1 dimensional singularities even under deformation? Concerning 3dimensional figures, those singularities would possibly be 1dimensional edges possibly connected to zerodimensional vertices.

November 17th, 2016, 04:51 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 150 Thanks: 64 Math Focus: Dynamical systems, analytic function theory, numerics 
What do you mean by singularities here? Does this mean the edges which generate trivial homology groups?

November 18th, 2016, 02:55 AM  #3 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 270 Thanks: 23 Math Focus: Number theory 
The best way I can describe a shape representative to my idea is a sphere transforming to a cube but barely differing from the former. Would singularities like edges and vertices be geometrically definable, even if the shape were just differentially different from a sphere? (Singularities here are places nondifferentiable in a at least one direction.) I now understand that for the shape being considered, under topology, genus zero would be conserved. If the consideration above is not one of topology, does it appear in math? 

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edges, preserving, topology, vertices 
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