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The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.

The Reidemeister move of type II.

The homeomorphic image of a so-called "complete separable" metric space. The continuous image of a Polish space is called a Souslin set.

A pseudo-Riemannian manifold is a manifold which has a metric that is of the signature diag(-,+,...,+), as compared to a Riemannian manifold, which has a signature of all ...

The Reeb foliation of the hypersphere S^3 is a foliation constructed as the union of two solid tori with common boundary.

The equivalence of manifolds under continuous deformation within the embedding space. Knots of opposite chirality have ambient isotopy, but not regular isotopy.

Eliminate each knot crossing by connecting each of the strands coming into the crossing to the adjacent strand leaving the crossing. The resulting strands no longer cross but ...

Two vector bundles are stably equivalent iff isomorphic vector bundles are obtained upon Whitney summing each vector bundle with a trivial vector bundle.

A topological space fulfilling the T0-separation axiom: For any two points x,y in X, there is an open set U such that x in U and y not in U or y in U and x not in U. ...

A topological space fulfilling the T3-separation axiom: X fulfils the T1-separation axiom and is regular. According to the terminology of Alexandroff and Hopf (1972), ...