Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.
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Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.
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Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on Rn (for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth).
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For a list of differential topology topics, see the following reference: List of differential geometry topics.
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Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.