• Discrete topology − All subsets are open.
• Indiscrete topology, chaotic topology, or Trivial topology − Only the empty set and its complement are … See more

• Alexandrov topology
• Lexicographic order topology on the unit square
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• Alexander horned sphere − A particular embedding of a sphere into 3-dimensional Euclidean space.
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• Auxiliary normed spaces
• Finest locally convex topology
• Finest vector topology See more

• Apollonian gasket
• Cantor set − A subset of the closed interval $${\displaystyle [0,1]}$$ with remarkable properties. See more

• Branching line − A non-Hausdorff manifold.
• Double origin topology
• E8 manifold − A topological manifold that does not admit a See more

The following topologies are a known source of counterexamples for point-set topology.
• Alexandroff plank
• Appert topology − A Hausdorff, perfectly normal (T6), zero-dimensional space that is countable, but neither See more